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HARVARD UNIVERSITY 




PHYSICS RESEARCH 
LIBRARY 

Gift of J. H. Van Vleck 




(^j^.^KwTl 





A COURSE OF 

MODERN ANALYSIS 



lonlion: C. J. OLAY and SONS, 
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 

AYE MARU LANE. 



•UuwoiD: 60, WELLINGTON STREET. 




lrtp}tg: F. A. BBOCKHAUS. 

^ciD «odt: THE MAOMILLAN GOMPANT. 

SombSB and ColnitU: MAOMILLAN AND 00.. Ltd. 



[All RighU reserved.] 



lS7T>A(.Z, 



A COURSE OF 



MODERN ANALYSIS 



AN INTRODUCTION TO THE GENERAL THEORY OF 

INFINITE SERIES AND OF ANALYTIC FUNCTIONS; 

WITH AN ACCOUNT OF THE PRINCIPAL 

TRANSCENDENTAL FUNCTIONS 



BY 



K T. WHITTAKER, M.A, 

FELLOW AND LECTURER OF TRINITY COLLEGE, CAMBRIDGE 



CAMBRIDGE: 
AT THE UNIVERSITY PRESS. 

1902 



loo 



Cambrttgr : 



PRINTBD BT J. AHD 0. F. CLAT, 
AT THE UNIVBR8ITT PRR88. 



Physics Research Librar> 

Jefferson Laboratoiy 

Harvard University 

MAR 19 1982 



-?"'- ' 




V 



y 



PREFACE. 



The first half -of this book contains an account of those methods and 
processes of higher mathematical analysis, which seem to be of greatest 
importance at the present time ; as will be seen by a glance at the table 
of contents, it is chiefly concerned with the properties of infinite series 
and complex integrals, and their applications to the analytical expression 
of functions. A discussion of infinite determinants and of asymptotic 
expansions has been included, as it seemed to be called for by the value of 
these theories in connexion with linear differential equations and astronomy. 

In the second half of the book, the methods of the earlier part are 
applied in order to furnish the theory of the principal functions of analysis — 
the Gamma, Legendre, Bessel, Hypergeometric, and Elliptic Functions. An 
account has also been given of those solutions of the partial differential 
equations of mathematical physics, which can be constructed by the help 
of these functions. 

My grateful thanks are due to two members of Trinity College, 
Rev. E. M. Radford, M.A. (now of St John's School, Leatherhead), and 
Mr J. E. Wright, B.A., who with great kindness and care have read the 
proof-sheets; and to Professor Forsyth, for many helpful consultations 
during the progress of the work. My great indebtedness to Dr Hobson*s 
memoirs on Legendre functions must be specially mentioned here; and 
I must thank the staff of. the University Press for their excellent co- 
operation in the production of the volume. 

E. T. WHITTAKER. 

Cambridge, 

1902 August 5 



CONTENTS. 



PART I. THE PROCESSES OF ANALYSIS. 



CHAPTER I. 

COMPLEX NUMBERS. 

8KCTI0N PAGE 

3 

4 



1. Heal numbers 

2. Complex uumbers 

3. The modulus of a complex quantity 5 

4. The geometrical interpretation of complex numbers 6 

Miscellaneous Examples 7 



CHAPTER II. 

THE THEORY OF ABSOLUTE CONVERGENCE. 

5. The limit of a sequence of quantities 8 

6. The necessary and sufficient conditions fi)r the existence of a limit . . 8 

7. Convergence of an infinite series 10 

8. Absolute convergence and semi-convergence 12 

9. The geometric series, and the series 2n~* 13 

10. The comparison-theorem 14 

11. Discussion of a special series of importance 16 

12. A convergency-test which depends on the ratio of the successive terms 

of a series 17 

13. A general theorem on those series for which Limit ( -* ) is 1 . . . 18 

n-ao \u^ J 

14. Convergence of the hypergeometric series 20 

15. Effect of changing the order of the terms in a series 21 

16. The fundamental property of absolutely convergent series .... 22 

17. Riemann's theorem on semi-convergent series 22 

18. Cauch/s theorem on the multiplication of absolutely convergent series . . 24 

19. Mertens' theorem on the multiplication of a semi-convergent series by an 

absolutely convergent series 25 

20. Abel's result on the multiplication of series 26 

21. Power-series 28 



X CONTENTS. 

SECTION PAGE 

22. Convergence of series derived from a power-series 30 

23. Infinite products 31 

24. Some examples of infinite products 32 

25. Cauchy's theorem on products which are not absolutely convergent . . 34 

26. Infinite determinants 35 

27. Convergence of an infinite determinant 36 

28. Persistence of convergence when the elements are changed .... 37 
Miscellaneous Examples 37 



CHAPTER III. 

THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS; 
TAYLOR'S, LAURENT'S, AND LIOUVILLE'S THEOREMS. 

29. The dependence of one complex number on another 40 

30. Continuity 41 

31. Definite integrals 42 

32. Limit to the value of a definite integral 44 

33. Property of the elementary functions 44 

34. Occasional failure of the property ; singularities 45 

35. The analytic function 45 

36. Cauchy's theorem on the integral of a Amotion round a contour . . 47 

37. The value of a function at a point, expressed as an integral taken round 

a contour enclosing the point 50 

38. The higher derivates 51 

39. Taylor's theorem 54 

40. Forms of the remainder in Taylor's series 56 

41. The process of continuation 57 

42. The identity of a function 59 

43. Laurent's theorem 60 

44. The nature of the singularities of a one-valued function .... 63 

45. The point at infinity 64 

46. Many-valued functions 66 

47. Liouville's theorem 69 

48. Functions with no essential singularities 69 

'Miscellaneous Examples 70 



CHAPTER IV. 



THE UNIFORM CONVERGENCE OF INFINITE SERIES. 

■ 

49. Uniform convergence 

50. Connexion of discontinuity with non-uniform convergence 

51. Distinction between absolute and uniform convergence 

52. Condition for uniform convergence 

53. Integration of infinite series . 

54. Differentiation of infinite series 

55. Uniform convergence of power-series 
Miscellaneous Examples 



73 
76 

77 
78 
78 
81 
81 
82 



CONTENTS. XI 

CHAPTER V. 

THE THEORY OF RESIDUES; APPLICATION TO THE EVALUATION 

OF REAL DEFINITE INTEGRALS. 

SECTION PAGE 

56. Residues 83 

57. Evaliiation of real definite integrals 84 

58. Evaluation of the definite integral of a rational function .... 91 

59. Cauchy's int^ral 92 

60. The number of roots of an equation contained within a contour . . 92 

61. Connexion between the zeros of a function and the zeros of its derivate . 93 

MiSCSLLAKBOUS EXAMPLES 94 

CHAPTER VI. 

THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 

62. Darboux's formula 96 

63. The Bemoullian numbers and the Bemoullian polynomials ... 97 

64. The Maclaurin-Bemoidlian expansion ........ 99 

65. Burmann's theorem 100 

66. Teixeira's extended form of Burmann's theorem 102 

67. Evaluation of the coefficients 103 

68. Expansion of a function of a root of an equation, in terms of a parameter 

occurring in the equation 105 

69. Lagrange's theorem 106 

70. Rouch^'s extension of Lagrange's theorem 108 

71. Teixeira*s generalisation of Lagrange's theorem 109 

72. Laplace's extension of Lagrange's theorem 109 

73. A further generalisation of Taylor's theorem 110 

74. The expansion of a function as a series of rational functions . . . Ill 

75. Expansion of a function as an infinite product 114 

76. Elxpansion of a periodic function as a series of cotangents . . . . 116 

77. Expansion in inverse factorials 117 

Miscellaneous Examples 119 

CHAPTER VII. 
FOURIER SERIES. 

78. Definition of Fourier series ; nature of the region within which a Fourier 

series converges 127 

79. Values of the coefficients in terms of the sum of a Fourier series, when the 

series converges at all points in a belt of finite breadth in the «-plane . . 130 

80. Fourier's theorem 131 

81. The representation of a function by Fourier series for ranges other than 

to 27r 137 

82. The sine and cosine series 138 

83. Alternative proof of Fourier's theorem 140 

84. Nature of the convergence of a Fourier series 147 

85. Determination of points of discontinuity 161 

86. The uniqueness of the Fourier expansion 162 

Miscellaneous Examples 167 



XU CONTENTS. 



CHAPTER VIII. 
ASYMPTOTIC EXPANSIONS. 

SECTION PAOK 

87. Simple example of an asymptotic expansion 163 

88. Definition of an asymptotic expansion 164 

89. Another example of an asymptotic expansion 165 

90. Multiplication of asymptotic expansions 167 

91. Integration of asymptotic expansions 168 

92. Uniqueness of an asymptotic expansion 168 

Miscellaneous Examples 169 



PART II. TRANSCENDENTAL FUNCTIONS. 



CHAPTER IX. 

THE GAMMA-FUNCTION. 

93. Definition of the Qamma-f unction, Euler's form 173 

94. The Weierstrassian form for the Gamma-function . . . . . 174 

95. The difference-equation satisfied by the Gamma-function . . . 176 

96. Evaluation of a general class of infinite products 177 

97. Connexion between the Gamma-fimction and the circular functions . 179 

98. The multiplication-theorem of Gauss and Legendre 179 

99. Expansions for the logarithmic derivates of the Gamma-function . . 180 

100. Heine*s expression of F («) as a contour-integral 181 

101. Expression of T(z) as a definite integral, whose path of integration 

is real 183 

102. Extension of the definite-integral expression to the case in which the 

argument of the Gamma-fimction is negative 184 

103. Gauss' expression of the logarithmic derivate of the Gamma-fimction as 

a definite integral 185 

104. Binet's expression of log F (e) in terms of a definite integral . . 186 

105. The Eulerian int^ral of the first kind 189 

106. Expression of the Eulerian integral of the first kind in terms of Gkunma- 

functions 190 

107. Evaluation of trigonometric integrals in terms of the Gamma-function 191 

108. Dirichlet's multiple integrals 191 

109. The asymptotic expansion of the logarithm of the Gamma-fiinction (Stirling's 

series) 193 

110. Asymptotic expansion of the Gamma-fimction 194 

Miscellaneous Examples 195 



CONTENXa xm 

CHAPTER X. 

LEGENDRE FUNCTIONS. 

BKCTION PAGE 

111. DefinitioD of Legendre polynomiab 204 

112. Schlafli's integral for P„ ^ 206 

113. Rodrigues' formula for the Legendre polynomials 206 

114. Legendre's differential equation 206 

115. The int^^ral-properties of the L^endre polynomials 207 

116. Legendre functions 208 

117. The recurrence-formulae 210 

118. Evaluation of the integral-expression for P^{z\ as a power-series . . 213 

119. Laplace's iut^pral-expression iot P^{z) 215 

120. The Mehler-Dirichlet definite integral for /*„(«). . % . . . 218 

121. Expansion of P^ {z) as a series of powers of 1/z 220 

122. The Legendre functions of the second kind 221 

123. Expansion of Q^isi) as a power-series 222 

124. The recurrence-formulae for the Legendre function of the second kind . . 2^4 

125. Laplace's int^;ral for the Legendre function of the second kind . . . 225 

126. Relation between Pm(^) ^^^ Q%{^\ when n is an integer . . . 226 

127. Expansion of {t-x)~^ as a series of Legendre polynomials . . 228 

128. Neumann's expansion of an arbitrary function as a series of Legendre 

polynomials 230 

129. The associated functions P^ (z) and $»"•(«) 231 

130. The definite integrals of the associated Legendre functions . . . 232 

131. Expansion of P%^{z) as a definite integral of Laplace's type . . 233 

132. Alternative expression of P^ {z) as a definite integral of Laplace's type . 234 

133. The function C/ («) 235 

MiSCBLLANBOUS EXAMPLES 236 



CHAPTER XI. 
HYPERGEOMETRIC FUNCTIONS. 

134. The hypergeometric series 240 

136. Value of the series /'(a, 6, c, 1) . 241 

136. The differential equation satisfied by the hypergeometric series 242 

137. The differential equation of the general hypergeometric function 242 

138. The L^^ndre functions as a particular case of the hypergeometric function . 245 

139. Transformations of the general hypergeometric function .... 246 

140. The twenty-four particular solutions of the hypergeometric differential 

equation 249 

141. Relations between the particular solutions of the hypergeometric differential 

equation 251 

142. Solution of the general hypergeometric differential equation by a definite 

int^;ral 253 

143. Determination of the integral which represents P(*) 257 

144. Evaluation of a double-contour int^;ral 259 

145. Relations between contiguous hypergeometric functions .... 260 

MiSCBLLANBOUS EXAMPLES 263 



; 



xiv CONTENTS. 



CHAPTER Xll. 

BESSEL FUNCTIONS. 

SECTION PAGE 

146. The Bessel coefficients 266 

147. Bessel's differential equation 268 

148. Bessel's equation as a case of the hjpergeometric equation . . 269 

149. The general solution of Bessel's equation by Bessel functions whose order is 

not necessarily an integer 272 

150. The recurrence-formulae for the Bessel functions 274 

151. Relation between two Bessel functions whose orders differ by an integer . 275 

152. The roots of Bessel functions 277 

153. Expression of the Bessel coefficients as trigonometric integrals . . 277 

154. Extension of the integral-formula to the caae in which n is not an integer . 279 

155. A second expression of J^^ (z) as a definite integral whose path of integration 

is real 282 

156. Hankel's definite-integral solution of Bessel's differential equation . . 283 

157. Expression of J^^ (2), for all values of n and «, by an integral of Hankel's type 284 

158. Bessel functions as a limiting case of Legendre functions .... 287 

159. Bessel functions whose order is half an odd integer 288 

160. Expression of J^ (z) in a form which furnishes an approximate value to J^^ (z) 

for large real positive values of z 289 

161. The asymptotic expansion of the Bessel functions 292 

162. The second solution of Bessel's equation when the order is an integer . . 294 

163. Neumann's expansion ; determination of the coefficients .... 299 
164 Proof of Neumann's expansion 300 

165. Schl5milch's expansion of an arbitrary functiou in terms of Bessel functions 

of order zero 302 

166. Tabulation of the Bessel functions 304 

Miscellaneous Examples 304 



CHAPTER XIIL 

APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 

167. Introduction : illustration of the general method 309 

168. Laplace's equation ; the general solution ; certain particular solutions . 311 

169. The series-solution of Laplace's equation 314 

170. Determination of a solution of Laplace's equation which satisfies given 

boundary-conditions 315 

171. Particular solutions of Laplace's equation which depend on Bessel functions 317 

172. Solution of the equation ^ -I- ^jj-hr=0 318 

92 ^ 92 7 32 y 

173. Solution of the equation g^ + ^j-h-^ -I- r=0 319 

Miscellaneous Examples 321 



CONTENTS. XV 



CHAPTER XIV. 
THE ELLIPTIC FUNCTION ^(z), 

8BCTI0N PAGE 

174. Introduction 322 

175. Definition of jf> («) 323 

176. Periodicity, and other properties, of ^(z) 324 

177. The period-parallelograms 324 

178. Expression of the function fp (z) by means of an integral .... 325 

179. The homogeneity of the function i> (z) 329 

180. The addition- theorem for the function ip{z) 329 

181. Another form of the addition theorem 332 

182. The roots Ci, eg, 63 333 

183. Addition of a half-period to the aigument o{ jp{z) 334 

181 Integration of (cw:* + 46a73+6c^ + 4ci:a7+e)-* 336 

185. Another solution of the integration-problem 336 

186. Uniformisation of curves of genus unity 338 

Miscellaneous Examples 340 



CHAPTER XV. 

THE ELLIPTIC FUNCTIONS sn z, en z, dn z, 

187. Construction of a doubly-periodic function with two simple poles in each 

period-parallelogram 342 

188. Expression of the function f{z) by means of an integral .... 343 

189. The function sn^ 345 

190. The functions cm and dn^ 346 

191. Expression of en z and dn z by means of integrals 347 

192. The addition-theorem for the function dnz 348 

193. The addition-theorems for the functions sn z and en ^ . . . 350 
191 The constant K 351 

195. The periodicity of the elliptic functions with respect to ^ . . 351 

196. The constant K' 352 

197. The periodicity of the elliptic functions with respect to K+iK' . . 353 

198. The periodicity of the elliptic functions with respect to tX' . . . 353 

199. The behaviour of the functions snr, en-?, dnz, at the point z=iK' . . 354 

200. (General description of the functions sn z, en 2, dn z 355 

201. A geometrical illustration of the functions snz, cnz, dnz. . . 356 

202. Connexion of the function snz with the function ^{z) . . . . 356 

203. Expansion of snz as a trigonometric series 357 

Miscellaneous Examples 359 



XVI CONTENTS. 



CHAPTER XVI. 

ELLIPTIC FUNCTIONS; GENERAL THEOREMS. 

SBCnON PAQK 

204. Relation betweeu the residues of an elliptic function .... 362 

205. The order of an elliptic Amotion 362 

206. Expression of any elliptic function in terms of fp {z) and fP' (z) . . 363 

207. Relation between any two elliptic fimctions which admit the same peri<xls . 364 

208. Relation between the zeros and poles of an elliptic function . . 365 

209. The function f (2) 366 

210. The quasi-periodicity of the function ((z) 367 

211. Expression of an elliptic function, when the principal part of its expansion 

at each of its singularities is given 367 

212. The function a{z) 368 

213. The quasi-periodicity of the function ^{z) 369 

214. The integration of an elliptic function 372 

215. Expression of an elliptic function whose zeros and poles are known . 372 
Miscellaneous Examples 374 

Index 377 



J 



PART I. 



THE PROCESSES OF ANALYSIS. 



W. A. 






CHAPTER I. 
Complex Numbers. 

1. Real Numbers. 

The idea of a set of numbers is derived in the first instance ifrom the 
consideration of the set of positive integral numbers, or positive integers ; 
that is to say, the numbers 1, 2, 3, 4, .... Positive integers have many 
properties, which will be found in treatises on the Theory of Integral 
Numbers; but at a very early stage in the development of mathematics 
it was found that they are inadequate to express all the quantities occurring 
in calculations ; and so this primitive number system has come to be 
enlarged. In elementary Arithmetic, and in the arithmetical applications 
of Algebra, several new classes of numbers are defined, namely rational 
fractions such as ^, negative numbers such as —3, and irrational numbers 
such as the number 1*414213..., which represents the square root of 2. 

The object of the introduction of these extended types of number is 
that we may express the result of performing the operations of addition, 
subtraction, multiplicatioo, division, involution, and evolution, on all integral 
numbers. Thus, the result of dividing the integer 1 by the integer 2 is 
inexpressible until we introduce the idea of fractional numbers: and the 
result of subtracting the integer 2 from the integer 1 is inexpressible until 
we introduce the idea of negative numbers. 

The totality of the numbers introduced up to this point is called the 
aggregate of real numbers. 

The extension of the idea of number, which has just been described, was not effected 
without some opposition from the more conservative mathematicians. In the latter half 
of the 18th centiuy, Maseres (1731—1824) and Frend (1767—1841) published works 
on Algebra, Trigonometry, etc., in which the use of negative quantities was disallowed, 
although Descartes had used them imrestrictedly more than a hundred years before. 

1—2 



4 THE PROCESSES OF ANALYSIS. [CHAP. L 

2. Complex Numbei^s*. 

If we attempt to perform the operations already named — multiplication, 
etc. — on any of the real numbers thus recognised, we find that there is one 
case in which the result of the operation cannot be expressed without the 
introduction of yet another type of numbers. The case referred to is that 
in which the operation of evolution is applied to a negative number, e.g. to 
find the square root of — 2. To express the results of this and similar opera- 
tions, we make use of a new number, deooted by the letter t; this is defined 
as a quantity which satisfies the fundamental laws of algebra (Le. can be 
combined with other numbers according to the associative, distributive, 
and commutative laws) and has for its square the negative number — 1. 

It is easily seen that all the quantities \\h\c\\ can be formed by com- 
bining i with real numbers are of the form a + 6t, where a and 6 are real 
numbers. A quantity a + 6i of this nature is called (after Gauss) a complex 
nv/mber. Real numbers may be regarded as a particular case of complex 
numbers, corresponding to a zero value of the quantity 6. 

The complex quantity thus introduced may in the first instance be 
regarded as formed by the association of the pair of real numbers a and 
6; as the quantities a, 6, i are subject to the ordinary laws of algebra, 
we obtain for the addition and multiplication of two complex numbers 
a + hi and c + dt the formulae 

(a + hi) + (c + di) = (o + c) +(6 + d)t, 

(a + 6t) (c + di) = (oc — bd) + {ad + he) i. 

But a complex number will usually be considered apart from its composition^ 
as an irresoluble entity. Regarded in this light, it satisfies the fundamental 
laws of algebra ; so that if a, 6, c are complex numbers, we have 

a + 6 a 6 + a, 

ah = 6a, 

{a-{'h)-{'C^a-{'{h-\'C\ 

ah ,c^a ,hc, 

a (6 + c) = oi + oc. 

It is found that the operations of multiplication, etc., when applied to 
complex numbers, do not lead to numbers of any fi"esh type ; the complex 
number will therefore for our purposes be taken as the most general type 
of number. 

The introduction of the complex number has led to many important 
developments in mathematics. Functions which, when real variables only 

* For the general theory of complex numbers, see Hankel, Theorie der eomplexen Zahlen- 
systeme (Leipzig, 1867), and Stolz, VarUsungen Uber allgemeine Arithmetik IL (Leipzig, 1886). 



2, 3] COMPLEX NUMBERS. 5 

are considered, appear as essentially distinct, are seen to be connected when 
complex variables are introduced : thus the circular functions are found to 
be expressible in terms of exponential functions of a complex argument, by 
the equations 

cos 07 = 2 («^ + e-^), 

sin a? = 2i (^** - ^"**)- 

Again, many of the most important theorems of modem analysis are 
not true if the quantities concerned are restricted to be real; thus, the 
theorem that every algebraic equation of degree n has n roots is true in 
general only when complex values of the roots are admitted. 

Hamilton's quaternions furnish an example of a still further extension of the idea 
of number. A quaternion 

is formed from four real numbers Wy x, y, z, and four number-units 1, i^ j, k, in the same 
way as the ordinary complex number .r+ty is formed from two real numbers ^, y, 
and two number-units 1, i Quaternions however do not obey the commutative law of 
multiplication. 

3. The modulus of a complex quantity. 

Let x+iy he & complex quantity; x and y being real numbers. Then 
the positive square root of a^ + y^ 18 called the modulus of (x + yi), and is 

written 

\x + yi\. 

Let us consider the complex number which is the sum of two known 
complex numbers, a? + iy and u + iv. We have 

(x + iy) + (u + iv) « (a? + w) + i (y + v). 
The modulus of the sum of the two numbers is therefore 

or {(aj« + y«) + (u» + v») + 2 (xu -h yv)]K 

But 

= (^ + y*) + (t*» + v") + 2 {(xu -^ yvY •¥ (xv - yu)«}*, 

and this latter expression is greater than (or at least equal to) 

(«" + y*) + (w' + i^)+2(xu + yv). 
We have therefore 

I r + iy I + 1 u + iy I > |(a7 + iy) + (u + iv) |, 

or the modulus of the sum of two complex numbers cannot be greater than the 
sum of their moduli; and in general it follows that the modulus of the sum 



6 THE PROCESSES OF ANALYSIS. [CHAP. I. 

of any number of complex quantities cannot be greater than the sum of their 
moduli. 

Let us consider next the complex number which is the product of two 
known complex numbers x -k-iy and u + %v; we have 

{x + iy) {u + iv) = {xu — yv) + 1 (aw + yu), 

and therefore 

I {x-\-iy){u + iv) I = [{xu - yvf + (a?w+ ye^)'}* 

^\x-^iy\ \u'\-iv\. 

The modulus of the prodiLCt of two complex quantities (and hence of any 
number of complex quantities) is therefore equal to the product of their modvli. 

4. The geometrical interpretation of complex numbers. 

For many purposes it is useful to represent complex numbers by a 
geometrical diagram, which may be done in the following way. 

Take rectangular axes Ox, Oy, in a plane. Then a point P whose 
coordinates referred to these axes are x, y, will be regarded as representing 
the complex number a?+ iy. In this way, to every point of the plane there 
corresponds some complex number; and conversely, to every possible complex 
number there corresponds one and only one point of the plane. 

The complex number x + iy may be denoted by a single letter z. The 
point P is then called the representative point or affix of the value z ; we 
shall also speak of the number z as being the affia of the point P. 

If we denote (a^+ y*)* by r and tan~* (^ i by d, then r and are clearly 

the radius vector and vectorial angle of the point P, referred to the origin 
and axis Ox, 

The representation of complex quantities thus afforded is often called the 
Argand diagram*. 

If Pi and Pa are the representative points corresponding to values Zi 
and z^ respectively of z, then the point which represents the value Zi-{- z^ is 
clearly the terminus of a line drawn fix)m Pi, equal and parallel to that 
which joins the origin to Pj. 

To find the point which represents the complex number z^z^, where Zx and 
z^ are two given complex' numbers, we notice that if 

z^ = ri (cos ^1 + i sin d,), 
iTj = r, (cos ^a + 1 sin d,), 

♦ J. R. Argand published it in 1806 ; it bad bowever previously been used by Gauss, and 
by Caspar Wessel, who discussed it in a miemoir published in 1797 to the Danish Academy. 



4] COMPLEX NUMBERS. 7 

then by multiplication 

z^Zi =s r^ra {cos (d, + 0^) + i sin (0i + ^,)}. 

The point which represents the value ZiZ^ has therefore a radius vector 
measured by the product of the radii vectores of Pi and Pa, and a vectorial 
angle equal to the sum of the vectorial angles of Pi and P,. 



Miscellaneous Examples. 

1. Shew that the representative points of the complex numbers 1+4^ 2+7i, 3 + lOt, 
areooUinear. 

2. Shew that a parabola can be drawn to pass through the representative points of 
the complex numbers 

2+t, 4+4i, 6 + 9t, 8+16t, 10+26i. 

3. Determine by aid of the Argand diagram the nth roots of unity ; and shew that the 
number of primitive roots (roots the powers of each of which give all the roots) is the 
number of integers including unity less than n and prime to it. 

Prove that if ^i, ^s* ^s* ••• ^ the argmnents of the primitive roots, 2coBpS=0 when 
p is A positive integer less than -j- — i , where a, b, c^„.k are the different constituent 

primes of n; and that, when p=-j- — .> 2co6jp^«» , , , where ii is the number of 

the constituent primes. 

(Cambridge Mathematical Tripos, Part I. 1895.) 



CHAPTER II. 
The Theory of Absolute Convergence. 

6. The limit of a sequence of quantities. 

Let Zi, z^i Zz, ... be a sequence of quantities (real or complex), ififinite in 
number. The sequence is said to tend towards a limiting value or limit I, 
provided that, corresponding to every positive quantity €, however small, a 
number n can be chosen, such that the inequality 

I Z^ - Z I < 6 

is true for all values of m greater than w. If £: is a variable quantity which 
takes in succession the values ^i, ^, ^t> ••• » then z is said to tend to the Ivmit L 

Example. Consider the sequence of numbers ^, ^, |)>-<9 for which ^=oii* This 
sequence tends to the limiting value 1=0; for if any positive quantity c be taken, and 
if n denote the integer next greater than - y^~o > ^^^^ ^^^ inequality 

1 

is true for all values of m greater than n. 

6. The necessary and sufficient condition for the existence of a limit 

We shall now shew that the necessary and sufficient condition for the 
existence of a limiting value of a sequence of finite numbers Zi, z^, ^,, ... is 
that corresponding to any given positive quantity e, however small, it shall be 
possible to find a number n such that the equation 

is verified for aXl positive integral values of p. This may be expressed in 
words by the statement that a finite variable quantity has a limit if and 
only if its oscillations have the limit zero ; it may be regarded as one of the 
fundamental theorems of analysis. 

First, we have to shew that this condition is necessaiy, i.e, that it is 
satisfied whenever a limit exists. Suppose then that a limit I exists ; then 



5, 6] THE THEORY OF ABSOLUTE CONVERGENCK 9 

(§ 5) corresponding to any positive quantity €, however small, a number n 
can be chosen such that 

and I Zn^ — Z I < ^ , for all values of p ; 

therefore 

I ^n+p — Zn\<\ (Zn+p — ~ (^n — I 

<\Zf^-l\ + \Zn-l\ 
€ € 

which shews the necessity of the condition 

I '^«-H> ■" -^n I < ^1 

and thus establishes the first half of the theorem. 

Secondly, we have to shew that this condition is sufficient, i.e. that if it 
is satisfied, then a limit exists. Suppose then that this condition is satis- 
fied Let 

Zr = a?r + tyr, 

where Xr and iyr are the real and imaginary parts of Zr. Then if 

I Zf^p — 2r„ I < €, 

we have | {x,^ - x^) + i(yn+i> -yn)\<^ 

and therefore x^ — €< Xn^ < ^n + €, 

and yn-€< yn+p < yn + e. 

Now the number n is determined by the quantity €, which can be assigned 
arbitrarily. Let ni> ^ ^i ^4> ••• be the numbers which correspond in this 

€ € € € 

way to the quantities ^, j, -, =^,.... Let w* be the least of the quantities 

Xn + e, Xn^ + 5, ^n, + 7 , ••• ^n» + o^ , SO that the quantities w©, v^, ti„ ... are a 
decreasing sequence ; and let v^ be the greatest of the quantities 

€ € € 

so that the quantities Vo, Vi, «,,... are an increasing sequence ; and clearly 



Then any of the numbers in the li-sequence is greater than any of the 
numbers in the v-sequence, since we have 

tir>Vr>Vt, if r > «, 

and tV > t/, > v„ if r < « ; 



^S^K 



10 THE PROCESSES OF ANALYSIS. [CHAP. IL 

and the difference u^ — v^ can be made as small as we please by increasing 
k. These two sequences u and v therefore uniquely define a real number 
(rational or irrational) ^, such that { is less than any Dumber in the 
iz-sequence and greater than any number in the v-sequence, and the 
differences Ut — ^ and ^—vt can be made as small as we please by increasing k. 

Then w* - f < w* - v* < ^^^ , 

so k«*-fl<km-ti*|+|t**-f|<2F:Y + 2^<2^- 

Moreover, by hypothesis, 

where p is any positive integer ; and so 

Since -^j—^ can be made as small as we wish by increasing k, this inequality 

shews that the sequence a^, aj^, a:,, ... tends to the limit f. Similarly the 
sequence ^i, ^at yt> ••• tends to a limit rj. 

Thus if T be any small positive quantity, it is possible to choose a number 
m such that for all values of r greater than m we have 

and therefore (xr - f )* + (yr — f y* < t^, 

or \Zr-l\<T, 

where Z = f + iff. 

This inequality shews that the sequence of quantities Zn z^, Zt, .*. tends to 
the limit I ; which establishes the required result, namely that the condition 
expressed is sufficient to ensure the existence of a limit. 

7. Convergence of cm infinite series. 

Let til, t^, t^s, ... tin h^ a series of numbers (real or complex). Let the 
sum 

be denoted by Sn* 

Then the infinite series 

is said to be convergent, or to converge to a sum 8, if the sequence of numbers 
Si, 82, 89, ... tends to a definite limit 8 aa n tends to infinity. In other 
cases, the infinite series is said to be divergent. When the series converges 
the quantity 8 — 8n, which is the sum of the series 

tifi+i + **fH-« "^ tf n+t + • • • , 



7] THE THEORY OF ABSOLUTE CONVERGENCE. 11 

is called the remainder after n terms, and is frequently denoted by the 
symbol Rn. 

The sum Un+i + Wn+i + ... + thn^ 

will be denoted by Sn,p- 

It follows at once, by combining the above definition with the results 
of the last paragraph, that the necessary and sufficient condition for the 
convergence of an infinite series is that Sn^p shall tend to the limit zero 
as n tends to infinity, whatever p is. 

Since tVM = >S^n,i) it follows as a peurticular case that ibn+i must tend to 
zero as n tends to infinity, — in other words, the terms of a convergent series 
must ultimately become indefinitely small. But this last condition, though 
necessary, is not sufficient in itself to ensure the convergence of the series, 
as appears from a study of the series 

In this series, 

or Sn^n>2' 

Therefore S = 1 + Si,x 4- iS^., + ^4,4 + fii.« + S^e^ie + ... 

^ •■••222 •••> 
which is clearly infinite ; the series is therefore divergent. 

Infinite series were used by Lord Brouncker in P/dl, Trans, 1668, and the expressions 
oonvergent and divergent were introduced by Gregory in the same year. But the great 
mathematicians of the 18th century used infinite aeries fredy without, for the most part, 
considering the question of their convergence. Thus Euler gave the sum of the series 



as zero, on the ground that 



...+p + ^ + j+l+«+a«+«»+ (a) 



^+^+^+... = -i- (6) 

1 ^* 



and 14. + 4....= ' (c). 

z sr z— 1 

The error of course arises from the fact that the series (6) converges only when | ^ | < 1, and the 
series (c) converges only when \z\>l,ao the series (a) does not converge for any value otz. 

The modem theory of convergence may be said to date from the publication of Gauss' 

DisquigitioneM circa seriem infinitam 1+?^+... in 1812, and Cauchy** Analyse Algdhrique 

in 1821. See Reifi^ Oeschichte der imendltchen Reihen (Tubingen, 1889). 



„ .-■ ^.■" ■ W\. . Jl'.' 



12 THE PROCESSES OF ANALYSIS. [CHAP. IL 

8. AbsoltUe convergence and semi-convergence. 
In order that the series 

(which we shall frequently denote by Xun), whose terms are supposed to be 
any complex quantities, may be convergent, it is sufficient, but not necessary, 
that the series S | z^ | shall be convergent. 

For we have 

I ^n,P I = I ^Wi + ^n+a + ... + ^n+p \ 

< I l^n+i I + I ^+9 I + ... 4- I Un+p I , 

and this last expression is inBnitely small, whatever p may be, when n is 
infinitely great, provided the series S 1 2^ | is convergent. 

Although this condition is sufficient to ensure the convergence of the 
series Xun, it is not necessary, i.e. the series Sun can converge even when 
the series 2 1 Wn | diverges. This may be seen by considering the series 

1 2 + 8 4+6 ••• + n +•••• 

This series is convergent ; for writing it in the form 

1.1.1. 

or 2 12 + 30+*"' 

we see that its sum is greater than ^ > And that the partial sum obtained by 

truncating the series after its 2nth term increases as n increases; on the 
other hand, by writing it in the form 

we see that the sum is less than 1, and that the partial sum obtained by 
truncating the series after its (2n + l)th term decreases as n increases. 

These partial sums must therefore tend to some limit between x and 1, and 
so the series converges. But the series of moduli is 

^2^3^4+'*'' 

which as already shewn is divergent. In this case therefore, the divergence 
of the series of moduli does not entail the divergence of the series itself. 

Series whose convergence is due to the convergence of the series formed 
by the moduli of their terms possess special properties of great importance, 
and are called absolutely convergent series. Series which though convergent 
are not absolutely convergent (i.e. the series themselves converge, but the 
series of moduli diverge) are said to be semi-convergent or conditionaUy 
convergent 



8, 9] THE THEORY OF ABSOLUTE CONVERGENCE. 18 

CO 1 

9. The geometrical series, and the series S — . 

The convergence of a particular series is in most cases investigated, not 
by the direct consideration of the sum Snp, but (as will appear from the 
following articles) by a comparison of the given series with some other series 
which is known to be convergent or divergent. We shall now investigate 
the convergence of two of the series which are most frequently used as 
standards for comparison. 

(1) The geometrical series. 
The geometrical series is defined to be the series 

l + irH--j' + ^ + 2^.... 

Considering the series of moduli 

we have for it iSfn.i,= kl'^' + l-^^l^+'H- ... +|«^|*^, 



or ^n.p=l^r^j^. 



Now if I -2: 1 < 1, then -= — r-K- is finite for all values of jp, while \z\^^^ tends 



1-1*. 

to zero as n tends to infinity. The series 

1+ U| + U|>+... 



is therefore convergent so long as 1 2: { < 1, and therefore the geometric series is 
ahsolutely convergent so long as \z\<l. 

When I ir I > 1, the terms of the geometric series do not tend to zero as n 
increases, and the series is therefore divergent. 

(2) 2%««ertC8 j;; + 2j + gi+^+g, + .... 

« 1 

Consider now the series 2 — , where s is any positive real quantity. 

We have 2^ + 3^ < 2"*'^ 2^^ ' 

11114 1 

4« "^ 5* "^ 6* "^ 7« ^ 4' "^ 4^1 ' 

and so on. Thus the sum of any number of terms of the series is less than 
the sum of the corresponding terms of the series 

1 _1 11 
l»-i "^ 2«-^ "*" 4«"* S*-^ ' 

1 J_ 1 1 

^^ p=5"*" 2«-i "^ 2*^"*> ■*"2»(»-i) ■^•••» 



14 THE PROCESSES OF ANALYSIS. [OHAP. II. 

and hence the convergence of this last series would involve that of the 
original series. But this last series is a geometrical series, and is therefore 
convergent if 

1 , 

that is, if 8> 1. 

The series ^ —i is therefore convergent if s>l; and since its terms 

n=l w* 
are all real and positive, they are equal to their own moduli, and so the series 
of moduli of the terms is convergent ; that is, the convergence is absolute. 

If 5 = 1, the series becomes 

which we have already shewn to be divergent; and when «»!, it is d fortiori 
divergent, since the etfect of diminishing « is to increase the terms of the 

• 1 . 

series. The series 2 — w therefore divergent if s^l. 

10. The Comparison-Theorem. 

We shall now shew that a series 

» 

wUl be absolutely convergent^ provided 1 1^ | w always less than C\vn\, where 
C is any finite number independent of n, and v^ is the nth term of another 
series which is known to be absolutely convergent. 

For we have under these conditions 

I Un+i i + I Un+t I + . . . + I Un+p I < C^ 1 1 ^n+i | + | Vn+3 I + • • • + | t^n-hp I } > 

where n and p are any integers. But since the series Svn is absolutely 
convergent, the series 2 | Vn | is convergent, and so 

tends to zero as n increases, whatever p may be. It follows therefore that 

tends to zero as n increases, whatever p may be, i.e. the series 2 | Wn | is 
convergent. The series 2t*n is therefore absolutely convergent. 

Corollary, A series will be absolutely convergent if the ratio of its 
terms, to the corresponding terms of a series which is known to be abso- 
lutely convergent, is always finite. 

Example 1. Shew that the series 

cos « + Si cos 2«H-si cos 3« + 7% COB 4« + . .. 
2' o' 4' 

is absolutely convergent for all real values of t. 



10] THE THEORY OF ABSOLUTE CONVERGENCE. 15 

For wheD z ia real, we have | cobtl? | ^ 1, and therefore | ^g i ^ "§ • Th® moduli of 



the terms of the given series are therefore less than, or at most equal to, the corresponding 
terms of the series 

1 11 

which by § 9 is absolutely conveigent The given series is therefore absolutely convergent. 



Example 2. Shew that the series 

1 1 



V . ••> 



where z^^(^l+^e^, (n=l, 2, 3, ...) 

is conveigent for all values of z^ except the values z^z^ z^^z^,*... 

The geometric representation of comjdez numbers is helpful in discussing a question of 
this kind. Let values of the complex number z be represented on a plane : then the values 
'i) Hi %)••• ^U ^*^i^™ & series of points which for large values of n lie very near the 
circumference of the circle whose centre is the origin and whose radius is unity: so 
that in &ct the whole circumference of this circle may be r^arded as composed of points 
included in the values z^^. 

For these special values z^ of r, the given series is clearly divergent, since the term 
becomes infinite when z=z,^. The series is therefore divergent at ^U points z 



situated on the circumference of the circle of radius unity. 

Suppose now that z has a value which is distinct from any of the values z^. Then 
is finite for all values of n, and less than some definite upper limit c : so the moduli 



of the terms of the given series are less than the corresponding terms of the series 

which is known to be absolutely convergent. The given series is therefore absolutely 
convergent for all values of r, except the values z^. 

It is interesting to notice that the area in the 2-plane over which the series converges 
is divided into two parts, between which there is no intercommunication, by the circle 

1*1-1. 

Example 3. Shew that the series 

2sin^+4sing+8sin — + ... + 2*sin ^-h.-- 

converges absolutely for all finite values of z. 
For when n is large, the quantity 

2,„ • WW 

"sm- 



2« 



3* 

has a value nearly unity; the given series is therefore absolutely convergent, since the 

comparison series 2 * ' is absolutely convergent. 



16 THE PROCESSES OF ANALYSIS. [CHAP. II. 

11. Disctission of a special aeries of importance. 

The theorem of § 10 enables us to establish the absolute convergence 
of a series which will be found to be of great importance in the theory of 
Elliptic Functions. 

Let Q>i and o), be any constants whose ratio is not purely real; and 
consider the series 



Us 



z^ 



>r 



{z — 2ma)i - 2nQ),)* (2ma>i + 2nft)jy 

where the summation extends over all positive and negative integral and 
zero values of m and n (the simultaneous zero values m = 0, n = excepted). 
At each of the points z = 2ma>i + 2nQ>s one term of the series is infinite, and 
the series therefore is not convergent. The absolute convergence of the 
series for all other values of z can be established as follows. 

Let z have any value not included in this set of exceptional values. 

The series may be written 



L + s ^ 

^ (2ma>i + 2n(o^ 



1- 



r-^- 



2mQ>i + 2nci>,> 

Now when | 2ma>i + ^na>^ \ is large (and we can suppose the series arranged 
in order of magnitude of | 2m6)i + 2m»2|), we have 

1 1 f r_i 

Limit o = ■!• 

2m(it)i + 27ica2 
The series is therefore absolutely convergent if the series 

2 ^ 



(2ma>i -♦- 271012)* 
is absolutely convergent : that is, if the series 

2 ^ - 

(2wicoi -f- 2r2G)a)' 
is absolutely convergent. 

To discuss the convergence of the latter series, let 

cDi = ai + t^Si , ©a = Oa + iySj, 

where aj, Oa, ySi, ^Sj, are real. Then the series of moduli of the terms of this 
series is 

This converges if the series 

2 •- (which we may denote by S) 

(m* 4- n^y 



■i 



11, 12] THE THEORY OF ABSOLUTE CONVERGENCE. IT 

converges ; for the quotient of corresponding terms is 

where M = - ; 

and this is never zero or infinite. 

We have therefore only to study the convergence of the series S. Now 



iit= -00 n= - 00 (m* + n^y 



00 00 1 

= 4 2 2 



where in the summation the occurrence of the pair of values m = 0, n = 
together is excluded. 

Separating S into the terms for which m = n, m>n, and m<n, re- 
spectively, we have 

00 1 00 yn-1 1 CO »-l 1 

i/Sf=2 -4-i+2 2 --^ — i+2 2 



m=i(2m')* w=i n=o (m'4-n*)* »-i »t=o (w' + ?i*)* 

»»-! 1 Wl 1 

But 2 . < i < — ;, . 



Therefore jfif< | -_^+ f 1+ i 1 



t=i 2*m' «=i ^ «=i ^ 



00 I ** 1 

But the series 2 —1 and 2 -„ are known to be convergent. So the 

series S is absolutely convergent. The original series is therefore absolutely 
convergent for all values of z except the specified excluded values. 

Example, Prove that the series 

1 
2 



{m^-\-m^-\- . . . 4-wir^r 

in which the summation extends over all positive and n^ative integral values and zero 
values of m,, m,, ... m,, except the set of simultaneous zero values, is absolutely convergent 

if yi>- . (Eisenstein, CrelUs Journal^ xxxv.) 

It 

12. A convergency-test which depends on the ratio of the successive terms 
of a series. 

We shall now shew that a series 

^1 + ^ + Us + ^4 + • • • 

is absolutely convergent, provided that for all valves of n greater than some 

w.A. 2 



18 THE PROCESSES OF ANALYSIS. [CHAP. II. 

fixed value r, the quantity \ — ^1 is less than K, where K is some positive quantity 

Un 

independent of n and less than unity. 
For the terms of the series 

are respectively less than the terms of the series 

which is a geometric series, and therefore absolutely convergent when K <1. 

Thus if —^ tends as n increases to a limiting value which is less than 
unity, the series is absolutely convergent. 

Example 1. If | c |< 1, shew that the series 

11=1 
converges absolutely for all values of z. 

For the ratio of the (n + l)th term to the nth is 

or c**+V, 

and if I c |<I, this is ultimately indefinitely smalL 

Example 2. Shew that the series 

,.«"fe^, (a-6)(a-2fe) (a-fe)(a-26)(a-36) 

converges absolutely so long as 1 2 |<-T-i . 

For the ratio of the (»+l)th term to the nth is — r ^> ^^ ultimately - 6z : so the con- 
dition for absolute convei^ence is 1 62; |<1, or U|<--t-: . 



Example 3. Shew that the series 2 converges absolutely so long as 

2|<1. 



For when |2|<1, the terms of the series bear a finite ratio to those of the series 
2 n2"~^; but th^ latter series is then absolutely convergent, since the ratio of the 



00 



(n+ l)th term to the nth is f 1 H — j z^ which tends to a limit less than unity as n increasea 



13. A general theorem on series /or which Limit 



»=oo 



^n+i 



= 1. 



It is obvious that if, for all values of n greater than some fixed value r. 



13] 



THE THEOBY OF ABSOLUTE CONVERGENCE. 



19 



t^n+i I is greater than \iLn\, then the terms of the series do not tend to zero as 



w« 



n+i 



U 



n 



n increases, and the series is therefore divergent. On the other hand, if 

is always leas than some quantity which is itself less than unity, we have 
shewn in § 12 that the series is absolutely convergent. The limiting case 



is that in which, as n increases, 



u 



n 



tends to the value unity. In this case 



a further investigation is necessary. 



We shall now shew that a series 

Wi + W2 + t^+ '", 



in which 



^+1 



tends to the limit unity as n increases, will he absolutely con- 



m 

vergent if, for ail values of n after some fixed value, we have 






<1- 



1 + c 



n 



where c is a positive quantity independent of n. 

For compare the series 2 | t/n | with the convergent series Svn* where 



A 
v«= - 



n 



i+, 



and -4 is a constant ; we have 



Vn Vn+l/ \ nJ 



-(-I) 



1+1 .11 

= 1 — -^— -I- terms in — , — , 
n n^ if 



V, 



As n increases, -""^^ will therefore tend to the limit 

Vn 



1- 



Hi 

n 



so that after some value of n we shall have 



^+1 

Wn 



Vn+i 



V 



n 



By a suitable choice of the constant A, we can therefore secure that for 
all values of n we shall have 

As Sv^ is convergent, 2 1 1^ | is therefore convergent, and so 2wn is abso- 
lutely convergent. 

2—2 



20 



THE PROCESSES OF ANALYSIS. 



[chap. n. 



Corollary. If 
form 



^+1 
^ 



can be expanded in descending powers of n in the 






where -4i, A^, -4,, ... are independent of n, then the series is absolutely 
convergent if -4i < — 1. 

This is easily seen to follow from the fact that when n is large the terms 

become unimportant in comparison with A^, 

14. Convergence of the hypergeometric series. 

The theorems which have been given may be illustrated by a discussion 
of the convergence of the hypergeometric series, 



a,b a(a + l)b(b+l) 
■^l.c "^ 1.2.c(c + l) 



a(a-H)(a-f 2)6(6-H)(6 + 2) 

1.2.3.c(c + l)(c+2) ■*■•••' 



which is generally denoted by F (a, 6, c, z). 

If c is a negative integer, all the terms after the (1 — c)th will be infinite ; 
and if either a or 6 is a negative integer the series will terminate at the 
(1 — a)th or (1 — 6)th term as the case may be. We shall suppose these 
cases set aside, so that a, 6, and c are assumed not to be negative integers. 

The ratio of the (n -♦- l)th term to the nth is 

Un+i (a -♦- M — 1) (6 4- n - 1) 
= z. 



Therefore 



Un 



^n+i 



w(c+w — 1) 



U. 



n 



1 + 



a- 1 



n 



1 + 



6-1 



n 



1 + 



c-1 
n 



As n tends to infinity, this tends to the limit \z\. We see therefore by § 12 
that the series is absolutely convergent when \z\<l, and divergent when 

\z\>h 



When 1^1=1, we have 



^+1 



1 + 



g-l 

n 



1 + 



6-1 



n 



l_£^^(- !)•_... 



n 



n" 



1 + - ^ -+ terms m -- , — , etc. 

n n^ n' 



Now a, 6, c are in the most general case supposed to be complex numbers. 



14, 15] THE THEORY OF ABSOLUTE CONVERGENCE. 21 

Let them be given in terms of their real and imaginary parts by the 

equations 

a = a' 4- ia!\ 

c = c' -f ic". 
Then (neglecting the terms in — , — , etc.) we have 

1% Iv 



^+1 



ajH6W;-H-i(a" + 6"-c") 



n 



= 1 + 



a' + h'-c'-W fa" 4 h" - c"\») * 



n 



) - (" ' „ 



a' + 6'-c'-l , .1 1 ^ 

= 1 H h terms m — , — , etc. 

By § 13, the condition for absolute convergence is 

a' + 6'~c'<0. 

Hence when \z\=^\,ihe condition for ike absolute convergence of the hyper- 
geometric aeries is that the real part of a + b — c shall be negative, 

16. Effect of changing the order of the terms in a series. 

In an ordinary sum the order of the terms is of no importance, and can 
be varied without affecting^the result of the addition. In an infinite series 
however this is no longer the case, as will appear from the following example. 

T^f 'C 1.1 1.1.1 1.1.1 1. 

and Sf=i -1+1-1+1-1 + 

and let 2n and Sn denote the sums of their first n terms. These infinite 
series are formed of the same terms, but the order of the terms is different. 

Then if A: be any positive integer, 

n * 11111 

But ^*-^*-' = 2/fc— l+2A-]k=2^T-2ifc- 

Similarly p,_, -p,_ = ^ - -^-_-^ . 

A series of equations like this can be formed, of which the last is 

Adding these, we have 

,111 1 c» 



22 THE PROCESSES OF ANALYSia [CHAP. II. 

Thus 2jt = So: + s Safc. 



2 

Making k indefinitely great, this gives 

an equation which shews that the eflfect of deranging the order of the terms 
in S has been an alteration in the value of its sum. 

Example, If in the series 

the order of the terms be altered, so that the ratio of the number of positive terms to the 

number of negative terms in S^ is ultimately a\ shew that the sum of the series will 

become log (2a). 

(Manning.) 

16. The fundamental property of absolutely convergent aeries. 

We shall now shew that the sum of an absolutely convergent series is not 
affected by changing in any manner the order in which the terms occur 

For let iS = t^i + 1^, 4- 1/^ + ^4 + . . . 

be an absolutely convergent series, and let /S' be a series formed by the same 
terms in a different order. 

Suppose that in order to include the first n terms of Sy it is necessary to 
take m terms of S\ So if k be any number greater than m, we have 

^k = 'Sin + terms of S whose suflBx is greater than n. 

Therefore 

I 'S'A;' — /S I < I /Sin — /Si j -♦- the sum of the moduli of a number of terms of S 

whose suffix is greater than n 

When n tends to infinity, | /?„ — ^ I tends to zero since the series 8 is con- 
vergent, and the sum 

tends to zero also, since the series is absolutely convergent. 

Thus I St' — /Si I tends to zero when k is indefinitely increased; which 
establishes the required result. 

17. Riemann*s theorem on semi-convergent series. 
We shall now shew that a semi-convergent series 

Wl + ^ + 1^ + ^4 + . . . , 

with real terms, may be made to converge to any desired real value, by suitably 
disposing ike order in which the terms occur. This property stands in sharp 
contradiction to that proved in the last article; an example of it was 
afforded by the result of §15. 



16, 17] THE THEORY OF ABSOLUTE CONVERGENCE. 28 

To establish the theorem, let the positive terms in the series be 

"^j '^> ^i>,i ••• > 
and let the negative terms be 

'^j*,* "^1 "^i*,! ••• • 
Then the series 

and -^i»,-^-^ii,- ... 

cannot be both convergent : for if they were, the original series would be 
absolutely convergent: one of them must therefore be divergent: and the 
other cannot be convergent, since in that case the original series would be 
divergent. It follows that the series 

tAp, + 1^, + tij^ + . . . 
and — t^, — ^n,-^— ... 

are both divergent. 

Now let S be any real number, and let it be desired to change the order 
of the terms in the original series, in such a way as to cause it to converge 
to the sum &. Suppose that a terms of the series 

have to be taken in order to obtain a sum greater than fif, so that 

Take now a number 6 of the terms of the series 
such as are required to make the sum 
less than & : so that 

Take next a number c of the terms of the series 

such as are required to make the sum 

greater than S ; and then take a number d of the terms of the series 

in such a way as to make the sum 

less than & again ; and so on. 

Proceeding in this way, we obtain a series whose sum at any stage of 



24 THE PROCESSES OF ANALYSIS. [OHAP. II. 

the process, differs from S by less than the last term included. But the 

terms of the series 

lii + ^,4- w,+ ... 

are ultimately indefinitely small, since the series is convergent; we can 
therefore in this way obtain a series 

whose sum differs from S by as little as we please ; and it consists of the 
terms of the original series, disposed in a different order. This establishes 
the result above stated. 

Corollary. If the terms of the original series are complex, they can be 
disposed in such an order as to give an arbitrarily assigned value to either 
the real or the imaginary part of the sum. 

18. Cauchys theorem on the multiplication of absolutely convergent series. 
We shall now shew that if two series 

and T= Vi + Vj + Vt 4- ... 

are absolutely convergent, then the series 

formed by the products of their termSy written in a/ay order, is ahsohitely con- 
vergenty and has for sum ST. 

Suppose that in order to include all the terms of the product 

it is necessary to take m terms of P ; and let k be any number greater 
than m. 

Then 

-P* = (^ + ^+ ... +^n)(Vi + ^2+ ••• +Vn) + terms t^aV/i in which either a or yS 

is greater than n, 

so I Pjt- Sri < I Sr,Tn'-ST\ 4- terms | w. 1 1 v^ |. 

Let {u-k-p) be the greatest suffix contained in these suffixes a and /8. 

Then 

|P,-OT:^|Snrn-«2'| + lhf„+,|4-...4-|^n+pll{iVi;4-...4-|t;n+p|} 

4- {i Wi I 4- ... 4- I Mn 1} (i V„+, I 4- ... 4- I Vn+p I}. 

Now when n tends to infinity, 

I ^n+i i 4- 1 Un+% j 4- ... 4- 1 Un^p \ tcuds to zero, 
and I Vn+i I 4- ... 4- 1 Vn^p \ tends to zero, 

while their coefficients tend to finite limits. 

Therefore | P^ — ST \ tends to zero, which proves the theorem. 



18, 19] THE THEORY OF ABSOLUTE CONVERGENCE. 25 

Example 1. Shew that the series obtained by multiplying the two seriee 

l+£ + ^+f' + ?! + 
2 2* 2' 2* ''' 

and l+- + -2+'5+ — > 

Z Z /u 

converges so long as the representative point of z lies in the ring-shaped r^on bounded 
by the circles l^jal and |z|a2. 

For the first series converges only when |«|<2, and the second only when |«|>1, and 
both must converge if the product is to converge. 

Example 2. Prove by multiplication of series that 

{cos S? cos 52 1 fTT* 2 /cos 2^ cos 4? . \) . cos a? . cos 5^ . 

For the coefficient of cos (2r-|-l) 2 in the product on the left-hand side of the equation is 

_jr« 1 ; 1 f 1 , 1 1 

9 (2r+l)« 3 *=, (Uf \{U - 2r- 1)« "^ (2it+2r4- 1)«J ' 



or 



or 



or 



or 



or 



9(2r+l)2 3(2r+l)«tli\V2it-2r-l 2/?/ "*" W 2Xr+2r+l/ J' 
^2 1 • r 2 2 4 11 



r_2 2 4 1 

t(2i-)2'*"(2it-l)2 (2;fc-2r-l)(2ir4-2r+l)J 



9(2r+l)« 3(2r+l)«fcfi l(2i-)2^(2it-l)2 (2;fc-2r- l)(2ir4-2r+l)J ^3(2r4-l)*' 
9(2r + l)2'*"3(2r-hl)* 3(2r+l)2V 22'^3«'^42'^"7 '*"3(2r+l)*' 



»r2 . 1 2 TfS 



9(2r+l)2^(2r+l)* 3(2r+l)« ' 6 ' 

1 



(2r+l)*' 
which gives the required result. 

19. Mertens' theorem on the multiplication of a semi-convergent series hy 
an absolutely convergent series. 

We shall now shew that if a series 

S = i^-fi^ + w, + ... 
is semi-convergent, and another series 

is absolutely convergent, then the series 

where pn = U^Vn + t^Vn-i + ... + UnV^, 

is convergent, and its sum is ST. 

For Pn = the sum of all terms UaVp in which a + /S < n + 1 

=^(Ui + 1U+,,.+Un)(Vi-{-V2+..,+Vn)-VjUn-Vi{Un'¥Unr^i)-... 



26 THB PROCESSES OF ANALYSIS. [CHAP. II. 

Therefore 

|Pn-/Sfri<|/Sf„r„-/gr| + |w„||t;,| + |t;,|K+un-,l + ... 

Now let k denote some number about half-way between 1 and n\ let € be 
the greatest of the quantities 

and let 7 be the greatest of the quantities 

i ^n + . . . + Wn-*-i 1 1 • • • ! 1^ + ^n-i + . . . + ti<| | . 

Then 

As n tends to infinity, € and { | y^^., | + . . . + 1 Vn 1 } are infinitesimal, while 
{|i^s| + ... +1 Vfc+a|} and 7 are finite. So every term on the right-hand side 
of the last equation is infinitesimal, and therefore in the limit 

P = /gT, 
which establishes the theorem. 

20. AbeVa resuU on th^ multiplication of aeries. 

We shall next prove a still more general theorem due to Abel*, which 
may be stated thus : 

Let two series 2 u^ and 2 Vn converge to the limits U and V respec- 

tively, and let the quantity 

be denoted by Wn. Then if the series 

converges at all, it converges to the sum UV, 

It will be noticed that none of the series considered need be absolutely 
convergent. 

We shall follow a method of proof due to Cesarof. 

Lemma I. If a set of quantities «i, 52> *8> ••• t^^ ^ ^ limit s, then 

. . 1 * 

Limit - 2 «i = «. 

f»=oo n ,=1 

For if € be any small positive number, we can find a number k such that 
the inequality 

\Sr — s\<€ 

is satisfied for all values of r greater than k. We have therefore 

2 n 1 * 1 *"** 1 * 

- 2 «< = - 2 «i + - 2 « + - 2 («f — 5). 
Ui^i ni=i n i^jc ni^jc 

• Crelle'B Journal, i. (1827). 

t Bulletin des Sciences math. (2) xiv. (1890). 



20] THE THEORY OP ABSOLUTE CX)NVERGENCE. 27 

Thus - 2 «i s . < - 2 ^< + - 2 5i - 5 1 

1 4 1 I n-k + l 

< - 2 \8i\-\ €. 

TV t==l W 

1 * 

Now make n infinitely great compared with k ; then - 2 \si\ tends to zero, 

^ f=i 

fl Jfc 4- 1 

and tends to unity, 



1 \ 
Limit - 2 «t- - fi < € ; 



and so 

and as e can be made as small as we please, this establishes the Lemma. 



Lemma II, If, as n increases indefinitely^ On and bn tend respectively 
to the limits a and b, then 

Limit -{aibn + a^bn-i + . . . -h Onfti) = ab. 

n=oo n 

To prove this, let v be the greatest integer contained in ^n. Then if e be 
any small positive number, we can take n so great that the inequality 

6^ - 6 I < 6 

holds so long as r > n-^p. 

Hence |ai(6n-6) + aa(6^i- 6) + ... + a^(6n-r+i — b)\ 

< €{|ai| + I Oal + ... + I a„|}. 

Hence Limit - \ Oi (bn— b) + a^(bn^i — b) + ... + a„(6,^_„4.i — 6) | 

n 



n=cc 



<€Limit-{|ai| + | 09! + ... + |a„|} 
91=00 n 

< € I a I , by Lemma I. 

The right-hand side of this inequality can be made as small as we 
please; hence 

Limit - {oi (bn — 6) + a, (b^^i — 6) + ... + a„ (^n-i^+i — b)] = 0, 
n=<» n 

or Limit - (ai6„ + a^bn-j -!-...+ a„6»«„+i) 

n=oo W 

= ^6 X Limit- (oi + a? + . . . + a^) 

y=oo ^ 

= ^a6, by Lemma I. 
Similarly 

Limit - (a^+ibn-y + a^ibn-v+j + ... 4- (tnbi) = JaJ. 

n=« ^ 



28 THE PROCESSES OF ANALYSIS. [CHAP. II. 

Adding the last two equations, we have 

Limit (Oibn + Ojin-i + . . • + Onbi) = oi, 

11=00 

which establishes Lemma II. 

Now let Wn denote the sum of the n first terms of the series 

considered in the above enunciation of Abel's result, we have 

where Un and F„ are used to denote the sums of the first n terms of the series 
U and V, From thiswe have 

and so by Lemma II. it follows that 

Limit -(Tri-fTrj+ ... + Trn)= ^F. 

1»=ao W 

But if the set of quantities TTj, TFj, ]F,, ... tend to a limit W, we have 
by Lemma I. 

Limit - ( TT, + TT, + . . . + F„) = W. 

Hence W=UV, 

which establishes Abel's result. 

Example 1. Shew that the series 

1-2+2-1 + 

is convergent, but that its square (formed by AbePs rule), 

2 

V2 
is divergent. 



■--Hfa-S-a-^)-" 



Example 2. If the convergent series 

0—1 2^ y 4r"^'*' 

be multiplied by itself, the terms of the product being arranged as iu AbePs result, shew 
that the resulting series is divergent if r ^ ^, but that it converges to the sum ^S^ when 
r<i. 

(Cauchy and Cajori.) 

21. Power-Series, 
A series of the type 

in which the quantities Oo, ch, a,, a, ... are independent of z, is called a series 
proceeding according to ascending powers of z, or briefly a power-series. 



21] 



THE THEORY OF ABSOLUTE CONVERGENCE. 



29 



We shall now shew that if a power-series converges for any value Zq of z, 
it will he absolutely convergent for all values of z whose representative points 
are within a cirde^ which passes through z^ and has its centre at the origin. 



00 



For if z be such a point, we have | -? | < | «o !• Now since ^a^zj^ converges, 



the quantity a^z^ must tend to zero as n increases indefinitely, and so 
we can write 

Jn 



anl = 



^p • • • • 



where €» tends to zero as n increases. Thus 

Ittol + loil |^| + |a«|U(^ + ... = €o + €i - 

Zy 

Now ultimately every term in the series on the right-hand side is less 
than the corresponding term in the convergent geometric series 



z 




z 


s 


Z 




+ e. 


— 


+ €, 




Zq 




Zq 




^0 



00 
n=0 



Z 

^0 



n 



the series is therefore convergent; and so the power-series is absolutely 
convergent, as the series of moduli of its terms is a convergent series ; 
which establishes the result stated. 

It follows from this that the area in the ^-plane over which a power- 
series converges must always be a circle ; for if the series converges for any 
point outside the particular circle which has just been found, we can (by 
taking this point as the point Zq) obtain a new and larger circle within which 
the series will converge. 

The circle in the ^r-plane which includes all the values of z for which 

the power-series 

ao + ttiZ + a22^ + a^z^ + ... 

converges, is called the circle of convergence of the series. The radius of 
the circle is called the radius of convergence. 

The radius of convergence of a power-series may be infinitely great; 
as happens for instance in the case of the series 

which represents the function sin z ; in this case the series converges for all 
finite values of z real or complex, ie. over the whole 2:-plane. 

On the other hand, the radius of convergence of a power-series may be 
infinitely small ; thus in the case of the series 

1 -h 1! ^ + 2! £:«-h 3! -2^ + 4! ^ + ..., 



we have 



u 



n-fi 



^n\z\. 



30 THE PROCESSES OF ANALYSIS. [CHAP. II. 

which, for all values of n after some fixed value, is greater than unity when 
z has any value different from zero. The series converges therefore only at 
the point -? = 0, and its circle of convergence is infinitely small. 

A power-series may or may not converge for points which are actually on 
the circumference of the circle ; thus the series 

z z^ 2? z* 

whose radius of convergence is unity, converges or diverges at the point 2: = 1 
according as « is greater or not greater than unity, as was seen in § 9. 

22. Convergency of series derived from a power-series. 

Let aQ + aiZ + a^z^ + a^z^-^- a4Z* + ... 

be a power-series, and consider the series 

Oi -{• 2aiZ -\' SotZ* + ia^z* -^ ..., 

which is obtained by differentiating the power-series term by term. We 
shall now shew that the derived series has the same circle of convergence as the 
original series. . 

For let -? be a point within the circle of convergence of the power -series ; 
and choose a positive quantity r, intermediate in value between | z \ and the 

radius of convergence. Then, since the series 2 an r^ converges absolutely, its 

n=0 

terms must decrease indefinitely as n increases; and it must therefore be 

possible to find a positive quantity M, independent of n, such that the 

inequality 

M 

is true for all values of n. 

Then the terms of the series 

in\an\\z\^-' 

are less than the corresponding terms of the series 

But in this series we have 

Un n r \ nj r ' 

which, for all values of n greater than some fixed value, is constantly less than 
unity ; this comparison-series therefore converges, and so the series 

in\an\\z\^' 



22, 23] THE THEORY OF ABSOLUTE CONVERGENCE. 31 

converges ; that is, the series 2 non^^^ converges absolutely for all points z 

«— 1 

00 

situated within the circle of convergence of the original series 2 On-s^, and the 
two series have the same circle of convergence. 

Similarly it can be shewn that the series 2 -^^^ , which is obtained by 
integrating the original power-series term by term, has the same circle of 

00 

convergence as 2 OnZ^K 

23. Infinite ProdiLcts. 

We proceed now to the consideration of another class of analytical ex- 
pressions, known as infinite products. 

Let l+Oi, l + Oj, l+Os, ... be an infinite set of quantities. If as 
n increases indefinitely, the product 

(H-ai)(l + aj)(l + a,)...(l +an) 

(which we may denote by 11^) tends to a definite limit other than zero, this 
is called the value of the infinite product 

n = (l+ai)(l+a3)(l +0,) ..., 

and the product is said to be convergent 

The product is often written n (1 -f- a«). 

If the value of the product is independent of the order in which the 
factors occur, the convergence of the product is said to be absolute. 

The condition for absolute convergence is given by the following theorem : 
in order that the infinite product 

(l-f-ai)(l+a,)(l + a,)... 

may he ahsolutely convei^gent, it is necessary and sufficient that the series 

' Oi -f- Oj + aj -h . . . 
should he ahsolutely convergent. 

For Iln = e^**'^^^'''^"*"***^"*'*^"^-'*"^**^^"*'*'^, 

so that n is absolutely convergent or not according as the series 

log(l + Oi) 4- log (1 + Og) -h log (1+ a,) + ... 

is absolutely convergent or not. But since log(l + o^) is nearly equal to ar 
when Or is small, the terms of this series always bear finite ratios to the 
corresponding terms of the series 

and so the absolute convergence of one series entails that of the other ; which 
establishes the result*. 

* A disooBsion of the convergence of infinite products, in which the results are derived 
withont making use of the logarithmic function, is given by Pringsheim, Math, Ann. xxxoi. 
pp. 119—164. 



^ 



32 THE PROCESSES OF ANALYSIS. [CHAP. II. 

ExcMiple, Shew that the infinite product 

sin z sin \z sin ^z sin \z 
z ' ^z ' '~^z ' iz 

is absolutely convergent for all values of z, 

. z 
sm- 

For when n is large, ^~~ is of the form 1 — ^ , where \^ is finite ; and the series 

n 

* X * 1 

2 -^ is absolutely convergent, as is seen on comparing it with 2 ,« . The infinite pro- 

11=1 n* n=i w 

duct is therefore absolutely convergent. 

24. Some examples of injmite products. 
Consider the infinite product 

m 

sm 21 
which represents the function . 

In order to find whether it is absolutely convergent, we must consider the 

* ^2 ^* * 1 

series 2 ^-r , or — 2 — ; this series is absolutely convergent, and so the 

product is absolutely convergent for all finite values of z. 
But now let this product be written in the form 

(-i)(>-J)('-i)('-4)-- 

The absolute convergence of this product depends on that of the series 

z z z z 

But this series is only semi-convergent, since its series of moduli 

z\ \z\ \z\ \z' 



+ — -!--- +'77-'+... 



7r TT 27r 27r 
is divergent. In this form therefore the infinite product is not absolutely 

convergent, i.e. if the order of the factors ( 1 ± - 1 is deranged there is 

a risk of altering the value of the product. 

Lastly, let the same product be written in the form 



M)'-H('n)«1l('-l^)«"}{('-a- 

in which each of the expressions 



1 + I e mir 



>■ 



24] THE THEORY OF ABSOLUTE CONVERGENCE. 33 

is counted as a single term of the infinite product. The absolute convergence 
of this product depends on that of the series 



or 



( 27r»'*'"V'^( 27r^'^-)"^r27r«.2«'^-'V"^(~27r^:T^'*'--j' 



and the absolute convergence of this series follows from that of the series 

The infinite product in this last form is therefore again absolutely 

convergent, the adjunction of the factors e *'" having changed the con- 
vergence from conditional to absolute. 

Example 1. Prove that n -[(1 ) «*[ is absolutely convergent for all values of 

Zj if c is a constant other than a negative integer. 

For the infinite product is absolutely convergent provided the series 

Le. if 2 < — ii^—+ terms m — _, —. etcV is, 

* 1 
and- on comparison with the convergent series 2 —^ , this is seen to be the case. 

Example 2. Shew that n jl — (l — ) «"*[ converges for all points z situated 
outside a circle whose centre is the origin and radius unity. 

For the infinite product is absolutely convergent provided the series 

n=2 \ nj 
is absolutely convergent. But as n increases, (1 — ) tends to the finite limit e, so the 



ratio of the (n + l)th term of the series to the nth term is ultimately - ; there is therefore 

z 

absolute convergence when , - 



<1, or |z|>l. 



Example 3. Shew that 



1.2.3...(w-l) 

— - -> - — ^- — -- n' 



z{z+l)iz + 2).„(z-\-n-\) 
tends to a finite limit as n increases indefinitely, unless « is a negative integer. 

W. A. 3 



34 THE PROCESSES OF ANALYSIS. [CHAP. II. 

For the expression can be regarded as a product of which the nth term is 

This product is therefore absolutely convergent, provided the series 

is absolutely convergent ; and a comparison with the convergent series 2 —^ shews that 

this is the case. When 2 is a negative integer the expression clearly becomes infinite owing 
to the vanishing of one of the factors in the denominator. 

Example 4. Prove that 

'(■-^)(-i)(-:)('-s)(>-4i)('-i>-'-'--^^ 

For the given product 

-■ir:"(-i)(-i)('-;)-{'-<M^)('-K)('-s) 

*\ ' 2^ 8 4^1 • 2fc-l 2*^*/ 



:= Limit 



xz 



(>-;)- ('-s)'*-('-4)-^- (■-«)'- 



=.Uimte'H^'lH-"'^2hi-k)z(l--^e'(l-{-^e'i(^ 
since the product whose factors are 

is ahsolviely convergent and so the order of its factors can be altered. 

Since log2 = l -Hi-Hi- - > 

this sliews that the given product is equal to 









e V wn.z. 



26. Cauchys theorem on products which are not absolutely convergent 

We shall now shew that if 

ai + aa + cis + a4+ ... 

is a semi-convergent sei-ies of real terms, then the infinite product 

(l+OiXl+OaXl + Oj)... 



26, 26] THE THEORY OF ABSOLUTE CONVERGENCE. 36 

converges (though not absolutely) or diverges {to the value zero), according 
as the series 

ai*+ aj'H- 03*+... 

is convergent or divergent. 

For the infinite product in question converges (though not absolutely) 
or diverges (to the value zero) according as the series 

log (1 + a,) + log (1 + Oa) + ... 

is semi-convergent or diverges to the value — 00 . 



n««oo 



Now since the series 2 On is convergent, the quantities On ultimately 



«=i 



diminish indefinitely, and therefore we can write 



a,» 



log (1 + On) = a„ - ^ (1 + €n), 

where |€n| tends to zero as n tends to infinity. 

»=• 
If the series 2 a^* diverges, it is clear therefore that the series 2 log(l + a^) 

n=l 

must diverge to the value — 00 ; if on the other hand the series 2 an* con- 

n=l 

n=sao 

verges, the series 2 log (1 + On) is convergent. From this the results relating 

n=l 

to the infinite product follow at once. 

26. Infinite Determinants. 

Infinite series and infinite products are not by any means the only known 
cases of infinite processes which can lead to convergent results. The re- 
searches of Mr G. W. Hill in the Lunar Theory* brought into notice the 
possibilities of infinite determinants. 

The actual investigation of the convergence is due not to Hill but to Poincare, Bull, de 
la Soc. MoUk. de France^ xiv. (1886), p. 87. We shall follow the exposition given by 
H. von Koch, Ada Math. xvi. (1892), p. 217. 

Let Aije (i. A: = — X , . . . + 00 ) be a doubly-infinite set of given numbers, and 
denote by 

the determinant formed of the quantities ili^ (i, A = — m — . . . + m) ; then if, 
for indefinitely increasing values of m. the quantity D^ has a determinate 
limit Z), we shall say that the infinite determinant 

is convergent and has a value D, In the case in which the limit D does not 
exist, the determinant in question will be said to be divergent. 

* Beprinted in Acta Maihematicay rm. pp. 1 — 86 (1886). 

3—2 



36 



THE PROCESSES OF ANALYSIS. 



[chap. II. 



The elements Au{i = — <x> ,,. -^ ao) are said to form the prindpcU diagonal 
of the determinant D ; the elements Aijc{k = oo .,, + oo) are said to form the 
line i ; and the elements -4ijt(t = — oo...-f-x) are said to form the column k. 
Any element A^ is called a diagonal or a non-diagonal element, according 

Sisi = k or i^ k. The element Aq^q is called the origin of the determinant. 

27. Convergence of an infinite determinant 

We shall now shew that an infinite determinant converges, provided the 
product of the diagonal elements converges absolutely and the sum of th^e non- 
diagonal elements converges absolutely. 

For let the diagonal elements of an infinite determinant D be denoted 
by 1 + aii(i = — X ... + 00 ), and let the non-diagonal elements be denoted 

by ttifc ( t > A?, , "* J , so that the determinant is 



. . . X *i- C&_i_i ^—10 ^ 11 ■ * * 

... ^0—1 X I ^^'^W ^01 • • • 

... ^—1 ^0 ■*■ « ^11 • • • 



Then since the series 



S lott 



i^k^-ao 



is convergent, the product 

00 / 00 

p= n (1+ 2 

is . 00 \ *•= - 00 

is convergent. 



a* 



I) 



Now form the products 



m 



= n ( 



1 + 2 a^ 



m / m 

»„= n (1+ 2 



O'ik 



> 



then if, in the expansion of P,„, certain terms are replaced by zero and 
certain other terms have their signs changed, we shall obtain Dm ; thus, to 
each term in the expansion of Dm there corresponds in the expansion of Pm 
a term of equal or greater modulus. Now Dm+p — Dm represents the sum of 
those terms in the determinant Dm+p which vanish when the quantities 
a{jt {t. A: = + (m + 1) ... ± (m +p)] are replaced by zero ; and to each of these 
terms there corresponds a term of equal or greater modulus in Pm+p — Pm* 
Hence I i)m^« — D^ I ^ P«i+ti - Pm- 



m+p 



As the quantities Pm, Pm+u ••• tend to a fixed limit, the quantities Dm, 
Dm+i, •.. will therefore tend to a fixed limit. This establishes the proposition. 



27, 28] THE THEORY OF ABSOLUTE CONVERGENCE. 37 

28. We shall now shew that a determinant, of the convergent form 
already considered, remains convergent when the elements of any line are 
replaced by any set of quantities whose moduli are all less than som£ fixed 
positive number. 

Replace, for example, the elements 

of the line by the quantities 
which satisfy the inequality 

I /*r I < M, 

where fi is a, positive number; and let the new values of 2)t» and D be 

denoted by D^ and D\ Moreover, denote by Pm and P' the products 

obtained in suppressing in Pm, and P the factor corresponding to the index 
zero ; we see that no term of D^ caai have a greater modulus than the cor- 
responding term in the expansion of /JbPm ] and consequently, reasoning as 
in the last article, we have 

which establishes the result stated. 

Example, Shew that the necessary and sufficient condition for the absolute conver- 
gence of the infinite determinant 

1 oj ... 

01 1 oj ... 

02 1 O3 ••• 

is that the series 

shall be absolutely convergent. (von Koch.) 



Miscellaneous Examples. 

1. Find the range of values of ;s for which the series 

28iIl2^-4 8in*2!+8sin«2-...+(-l)* + l2*sin*•2+... 
is convergent. 

2. Shew that the series 

1 _ Jl^ _1 1_ 

is semi-convergent, except for certain exceptional values of z ; but that the series 

1+ L+...+ _L_ \ L__ ___L_+_L_+... 

t z-^-X '" z-\-p^l z-k-p z-^-p-^-l '" z-h2p-^q-\ z+2p'\-q "*' 
in which (p-^q) negative terms always follow p positive terms, is divergent. (Simon.) 



38 



THE PROCESSES OF ANALYSIS. 



[chap. II. 



3. Shew that the series 



1* 2^ 3* 4^ 



(l<a<i3) 



is oouvergent 

4. Shew that the series 

is convergent. 

5. Shew that the series 



(Cesaro.) 



a+i3«+a3+/3*+... 



(0<aO<l) 



(Cesaro.) 



7ig*-i 



2 



((-3"-'} 



2kiw 
m 



converges absolutely for all values of «, except the values 

(a=0, 1; k=0, 1, ...m-1 ; m«=l, 2, ... x). 

6. If 8^ denote the sum of the first n terms of a convergent series whose sum is «, 
shew that 

7. In the series whose general term is 

u^^qn-Pa: ^ , (0<q<l<x) 

where y denotes the number of figures in the expression of n in the ordinary decimal scale 

of notation, shew that 

i_ 

Limit tin* =y, 

and that the series is convergent, although the quantity ^^^^^ is infinitely great when n is 
infinitely great and of the form 1 + lO"- *. (Lerch.) 

8. Shew that the series 

4 

where qn=q'^~n, {0<q<l) 

is convergent, although the ratio of the (n + l)th term to the nth is greater than unity 
when n is not a triangular number. 

9. Shew that the series 



(Cesaro.) 



00 ^wix 



where w is real, and where {w-^-ny is understood to mean c*!**^***), the logarithm being 
taken in its arithmetic sense, is convergent for all values of «, when the imaginary part of 
X is positive, and is convergent for values of s whose real part is positive, when x is real. 

• (— l)n+l 

10. Shew that the qth power of the convergent series 2 - ;. is convergent when 

ll=sl w 



^ — <r, and divergent when ^ — >r. 
9 9 



(Cajori.) 



mSC. EXS.] THE THEORY OF ABSOLUTE CONVERGENCE. 



39 



11. If the two semi-convergent series 

i i^y^ and .<^i|^', 

where r and s lie between and 1, be multiplied together, and the product arranged as in 
Abel's result, shew that the necessary and sufi&cient condition for the convergence of the 
resulting series is r+«>l. (Cajori.) 

12. Shew that if the series 

be multiplied by itself any number of times, the terms of the product being arranged as 
in Abel's result, the resulting series converges. (Cajori.) 

13. Shew that the qth power of the series 

Oj sin ^+02 sin 2^+ . .. +a« sin n^-f • .. 

is convergent whenever ^ <r, r being the maximum niunber satisfying the relation 

for all values of n. 

14. Shew that if $ is not equal to or a multiple of 27r, and if the quantities 
Kq, u^, t£„ ... are all of the same sign and continually .diminish in such a way that the 
limit of tfM is zero when n is infinite, then the series Stt^ cos {n6+a) is convergent 

Shew also that, if the limit of t^ is not zero, but all the other conditions above are 

B , 6 . . 

satisfied, the sum of the series is oscillatory if - is commensurable, but that, if - is in- 

commensurable, the siun may have any value between certain limits whose difference is 
a cosec^^ where a is the limit of u^^ when n is infinite. 

(Cambridge Mathematical Tripos, 1896, Part I.) 

15. Prove that 



i{('-0 



.*-i 



s> 



+ »*-^^+.+ 



^1 



}■ 



where k is any positive integer, converges absolutely for all finite complex values of z. 
16. Let 2 ^M be an absolutely convergent series. Shew that the infinite determinant 



A(C) = 



(c-4)»-(?o -<?, 



-6, 







4?-e, 






(c-2)»-<»o 
2»-<?o' 

-61 



2»-tf, 
4»-(9. 



4»-<?<, 
2«-tf« 



-0. 
4«-tfo 

0»-A 



-6y (c+2)»-Oo 



4»-A 



4«-(?„ 



-^4 

2»-tfo 
(c + 4)»-^o 



converges : and shew that the equation 



ia equivalent to the equation 



A(c)-0 



sin' Jwc — A (0) sin* ^ir^o • 



(HilL) 



CHAPTER III. 

The Fundamental Properties of Analytic Functions ; 
Taylor's, Laurent's, and Liouville's Theorems. 

29. The dependence of one complex number on another. 

The problems with which Analysis is mainly occupied relate to the 
dependence of one complex number on another. If z and f are two complex 
numbers, so connected that the value of one of them is determined by the 
value of the other, e.g. if f is the square of z, then the two numbers are 
said to depend on each other. 

This dependence must not be confused with the most important case of 
it, which will be explained later under the title of analytic functionality. 

If ^ is a real function of a real variable z^ then the relation between ( and z, which 
may be written 

f=/(«), 

can be visualised by a curve in a plane, namely the locus of a point whose coordinates 
referred to rectangular axes in the plane are (z, C), No such simple and convenient 
geometrical figure can be found for the purpose of visualising an equation 

considered as defining the dependence of one complex number f=f +ii; on another 
complex number z=x-\-iy. A representation strictly analogous to the one already given 
for real variables would require four-dimensional space, since the niunber of quantities 
f > »7i ^1 y? is now four. 

One suggestion (made by Lie and Weierstrass) is to use a doubly-manifold systeln of 
lines in the quadruply-manifold totality of lines in three-dimensional space. 

Another suggestion is to represent £ and 17 separately by means of surfaces 

A third suggestion, due to Heffter*, is to write 

then draw the surface r^r{x,y) — which may be called the modular-surface of the 
function — and on it to express the values of B by surface-markings. It might be 
possible to modify this suggestion in various ways by representing B by curves drawn 
on the surface r=^r{x, y). 

* ZeitschriftfUr Math. u. Phys. xliv. (1899), p. 236. 



29, 30] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 41 

30. Continuity, 

Let f(z) be a quantity which, for all values of z lying within given 
limits, depends on z. 

Let Zi be a point situated within these limits. Then f(z) is said to be 
continuous at the point -^i, if, corresponding to any given positive quantity €, 
however small, a finite positive quantity 17 can be found, such that the 
inequality 

i/w-/(*,)i<« 

is satisfied so long as | ^ — ^1 { is less than 1;. 

U f(z) is continuous at z = Zi, and if its real and imaginary parts be 
denoted by u and v, then u and v depend continuously on z. 

For if f(z) = w + iv, we have 

I (u - 1^) + i (v - Vi) I < 6, 

and so {u — v^y -f (v — v^Y < €*, 

which gives {u — v^y < e^ and {v — ViY < e^, 

and so | u — -Mj | < e and \v — Vi\<€. 

The popular idea of continuity, so far aa it relates to a real variable ( depepding on 
another real variable z, is somewhat different to that just considered, and may perhaps 
best be expressed by the definition " The quantity f is said to depend continuously on z 
H as z passes through the series of all values intermediate between any two adjacent 
values Zi and z^, ( p&sses through the series of all values intermediate between the 
corresponding values d and fg-" 

The question thus arises, how far this popular definition is equivalent to the analytical 
definition given above. 

Cauchy shewed that if a real variable (, depending on a real quantity «, satisfies the 
analytical definition, then it also satisfies what we have called the popular definition. 
But the converse of this is not true, as was shewn by Darboux. This fact may be illus- 
trated by the following example*. 

Let B(^) denote the integer next less than x ; and let 



f(s)^.[l-E |^-^-^}]+^{j^} si 



IT 

sm^^ 



At a;=0, we have/(^)=»0. 

Between 4?= - 1 and 47= + 1 (except at a;=0), we have 

/(;r)=sin£. 



From this it is easily seen that/(^) depends continuously on x near a;=:0, in the sense 
of the popular definition, but is not continuous in the sense of the analytical definition. 

* Dae to MansioD, Mathens^ ix. (1899). 



42 THE PROCESSES OF ANALYSIS. [CHAP. III. 

31. Definite integrals. 

Let Zo and Z be any two values of z ; and let their representative points 
A and B in the ^-plane be connected by an arc (straight or curved) AB; 
and let Zi, z^, Zg, ... Zn be a number of points taken on the line AB in any 
manner. 

Let f(z) be a quantity which, for variations of z along the arc AB, 
depends continuously on z. 

Let Zq be any point situated in the interval ZqZi of the curve : let Zi be 
/ any point situated in the^||ntervap^i2:,: and so on: and consider the sum 

S ^f{z^){z, - Z,)'^f{z^){z, - -^0 + ... +/(0(^ - ^n). 

We shall shew that if the number n increases indefinitely^ in such a way 
that each of the quantities \ z^ — -?r-i I tenis to zero, then this sum wiU tend to 
afia>ed limit, independently of the way in which the points 

Z\i Z^y ••• Zn, Zq , Zi f ... Zn t 

are chosen. 

For let € be a given small positive quantity. Since f{z) is continuous, 
for each point z=^a of the arc AB we can find a quantity 7}a such that 

\f(z) -/(a) I < e. 

80 long as \z — a\<rfa' 

Let 7} be the least value of rja corresponding to points a on the arc AB. 
We shall suppose the subdivision of the arc has been carried so far that 
each quantity Ur — '^r-i| is less than tf, and shall first find the effect of 
putting in further subdivisions. 

Suppose then that the interval ZqZi is subdivided at points z^, z^, "*z^^\ 
that the interval z^z^ is subdivided at the points ^h> '^i9» ••• Zir^\ and so on : 
so that the sum s becomes 

«' =f(Zo")(Zoi - Zo) -^f(Zoi)(Zoi - ^oi) + . . . 
+/(0(^ii - ^i) +f{Zu){z,^ - ^ii) + . . . 
i" • • • » 

where z^' is any point in the interval z^z^, z^( is any point in the interval 
ZnZoi, and so on. 

Then 

s-s^ {f(zn ^f(Zo')] (^01 - ^o) + {f(zoO -/(V)! (^« - -^0,) + . . . 

+ l/(^i") -fM] (^n - Z,) + {/(^„') -/(^/)j {z,,^Zu) + ... 
^r • • . . 



31] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 48 

Therefore 

|«'-«| <€ {|^W-^o| + |'2oJ-'^Ol|+ •••} 

< € X the length of the broken line connecting the points -e^o» % i -^^w* • • • 

where I is the length of the arc AB, 

Now by making € indefinitely small, we can make the right-hand side of 
this equation as small as we please; and therefore the sum s tends to a 
definite limit when the number of subdivisions is indefinitely increased, 
provided that at each change in the subdivisions the old points of division 
are retained. 

The restriction contained in the last phrase has still to be removed. 
To do this, suppose that two different methods of division, in eaxih of which 
the quantities \Zr — Zr^i\ are less thfin 17, furnish sums s^ and «9. Now 
combine the two methods of division, so that every point of division in 
either of the original sjchemes becomes a point of division in the new 
scheme. Let the sum corresponding to this new method of division be Sj^, 
Then since by the above 

I «i — «ia I < €/ and I «s — «u I < ely 
we have I «i — *a I < 2€Z, 

which shews that «i and $2 tend to the same limit. The theorem is thus 
established. 

The limit thus shewn to exist is called the definite integral of f{z), 
taken along the arc AB\ it is denoted by 



I 



/: 



f{z)dz^ 

AB 

in cases where there is no ambiguity as to path, it may be denoted by 

f(z) dz. 

As an example* of the evaluation of a definite int^;ral directly from the definition, 
suppose it is required to find the definite integral of the continuously dependent quantity 

(1 -«*)"*, taken along the straight line (part of the real axis) joining the origin (««0) 
to a point z=Z^ where Z is real Denote the definite int^;ral by /. Then by definition, 

/= Limit 2 ^^'"^^ 

and the mode of choosing the points z^ and V is arbitrary, within the limits already 

explained ; we shall take 

«,i=sinrd, 

V = sin(r+i)d, 

where d= r sin"* Z. 

n+l 

• Netto, Zeitschrift/Ur Math. xl. (1896). 



44 THE PROCESSES OF ANALYSIS. [CHAP. III. 

Thus /-Limit I ^^^(r+l)b-^smrb 

n ^ 

= Limit 2 ^ sin 3 
= Limit 2 (n+ 1) sin 5 

. b 

sm- 
= sin ~^^ Limit 1 

2 
=8in-iZ. 

The value of the definite integral is therefore sin"^ Z, 

32. Limit to the value of a definite integral. 

Let M be the greatest value of !/(^)| at points on the arc of inte- 
gration AB, 

Then |/(V) {zi - z,) -\-f{z^) {z, - ^,) + • • • +/( V) {Z - Zn) \ 

<|/(V)l|^i--^o| + |/(Olk2--^i|+...+|/(V)|l^-^n| 
^ ilf {I ^1 - £-0 I -h i ^, - -?i I -h ... + I ^- -^n '} 

where I is the length of the arc of integration AB. 
We see therefore, on proceeding to the limit, that 

f{z)dz 

AB 

cannot be greater than the quantity ML 

33. Property of the elementary functions. 

The reader will be already familiar with the word function^ as used 
(in text-books on Algebra, Trigonometry, and the Differential Calculus) to 
denote analytical expressions depending on a variable z ; such for example as 

z^, c*, \ogz, svxr^z^. 

These quantities, formed by combinations of the elementary functions of 
analysis, have in common a remarkable property, which will now be investi- 
gated. 

Take as an example the function e^. 

Write e* =f{z). 

Then if z' be a point near the point z, we have 

z — z z — z z - z 



I 



( z-/ (z-zY ) 



32 — 35] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 45 

and benoe, if the point z tends to coincide with z, the limiting value of the 
quotient 

z' -z 
is (F. 

This shews that ^e limiting value of 

f{z'y^f{z) 

z'^z 

%8 in this case independent of the direction of the short path by which the 
point / 7noves towards coincidence with z, i.e. it is independent of the 
direction in which / lies as viewed from z. 

It will be found that this property is shared by all the well-known 
elementary functions ; namely, that if f{z) be one of these functions and 
h be any small complex quantity, the limiting value of 

Jl/(^ + A)-/(^)} 
is independent of the mode in which h tends to zero. 

34. Occasional failure of the property. 

For each of the elementary functions, however, there will be certain 
points z at which this property will cease to hold good. Thus it does not 

hold for the function at the point z = a, since the limiting value of 

z ^ a 

If 1^ 1^ 

h z — a — h z — a] 

is not finite when z=^a. Similarly it does not hold for the functions log^ 
and z^ at the point z = 0. 

These exceptional points are called singular points or singularities of the 
function f(z) under consideration ; at other points the function is said to be 
regular, 

36. The analytical function. 

The property noted in § 33 will be taken as the basis of our definition of 
an analytic function, which may be stated as follows. 

Let an area in the -e-plane be given ; and let w be a quantity which has 
a definite finite value corresponding to every point z in that area. Let 
z, z-^-izhe values of the variable z at two neighbouring points, and UyU-\rhu 
the corresponding values of u. Then if at every point z within the area 

^ tends to a finite limiting value when hz tends to zero, independently of 



46 THE PROCESSES OF ANALYSIS. [CHAP. Uh 

the way in which Bz tends to zero, u is said to be an analytic function of z, 
regular within the area. 

We shall generally use the word " function " alone to denote an analytic 
function, as the functions studied in this work will be almost exclusively 
analytic functions. 

In the foregoing definition, the function u has been defined only within 
a certain area in the ^-plane. As will be seen subsequently, however, the 
function u will generally exist for other values of z not excluded in this area : 
and (as in the case of the elementary functions already discussed) may have 
Angularities, for which the fundamental property no longer holds, at certain 
points outside the limits of the area. 

The definition of functionality must now be translated into analytical 
language. 

If /(^) be a function of z, regular in the neighbourhood of a particular 
value Zy then, by the definition, the quantity 

z — ^ 

tends to a definite limit, depending only on z, when / tends to z. Let this 
limit be denoted by the symbol /' (z). 

Then (by the definition of a limit) for every positive quantity 6, however 
small, it is possible to find a quantity 17, such that 

is less than €, so long as \z — z\ is less than 17. 
If therefore we write 

/(/) =f{z) + (/ - Z)f' {Z) + 6 (/ - Z\ 

we see that | e' | is less than e, so long as | / — ^ | is less than 17 ; that is, the 
function /(-gr) must be such that the quantity e', defined by the equation 

/(/) =/(^) + (/ - z)f (^) + « (/ - z\ 

tends to the limit zero as z tends to z. 

The necessity for a strict definition of the term "function" may be seen from the 
following consideration. 

Let y denote the temperature at a certain place at time U As t varies, y will vary, 
and y may loosely be called a "function" of t But y cannot be expressed in terms of t 
by a Maclaurin's infinite series 



'-w.-.-^'(l),./^(S), 



I • • • > 

** \w*V<-0 



for if it could, the knowledge of the temperature for a single day would enable us to 
determine the quantities 



36j THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 47 

and then from the Maclaurin's expansion it would be possible to predict the temperature 
for the future ! 

Maclaurin's series is in fact, as will appear subsequently, applicable only to analytic 
functions, in the sense in which analytic functions have been defined above. 

36. Cauchys theorem on the integral of a function round a contour. 

A simple closed curve in the plane of the variable z is often called 
a contour : if -4, jB, C, D be points taken in order along the arc of the 
contour, and i{f(z) be a quantity depending on z and continuous at all points 
on the arc, then the integral 



/ 



f(z)dz, 

ABCDA 

taken round the contour, starting from the point A and returning to A again, 
is called the integral of the quantity f(z) taken round the contour. Clearly 
the value of the integral taken round the contour is unaltered if some point 
in the contour other than A is taken as the starting-point. 

We shall now prove a result due to Cauchy, which may be stated as 
foUowa If f(z)i8 an analytic function, regular at all points in the interior 
of a contour, then 



I 



Az)dz=0, 

where the integration is taken round the contour. 

For let A, B, (7, D be points in order on the contour. Join A to C hy an 
arc AECy which will divide the region contained within the contour into two 
distinct portions. Then the integral taken round the contour ABCDA is 
equal to the sum of the integrals taken round the two contours ABCEA and 
ABCDA ; for 



f f{z)dz+{ f{z)dz 

J ABCEA J ABCDA 



= f f{^)dz+j f(z)dz+f f(z)dz-^j f(z)dz 

J ABC J CEA J ABC J CD A 

= f f{z) dz. 

J ABCDA 

since the integrals along CEA'^ and AEC neutralise each other. 

Now join any point E on the arc AEC to D by an arc EFD, and join 
^ to jS by an arc EOB; then in the same way we see that the integral 
round ABCEA is equal to the sum of the integrals round ABOEA and 
EOBCE, and the integral round AECDA is equal to the sum of the 
integrals round AEFDA and DFECD. 

Thus the original contour-integral is equal to the sum of the integrals 



48 THE PROCESSES OF ANALYSIS. [CHAP. III. 

round the four contours ABGEA, EGBCE, AEFDA, DFECD, into which it 
has been divided by drawing the cross-lines. 

Proceeding in this way by drawing more cros^-lines, we see that the 
original contour-integral can be decomposed into the sum of any number of 
integrals round smaller contours, which constitute a network filling up the 
original contour. 

Now suppose that each of these small contours has linear dimensions of 
the same order of magnitude as a small quantity l. Let z^^ be a point within 
one of them. Then on this small contour we have 

f{z) ^f(Zo) + (Z- Zo)f (Z,) + (^ - Zo) €, 

where e is infinitely small when I is infinitely small. 

Thus ^f{z)dz^^f{z,) dz-^-jiz - Zo)f{z,) dz+j{z--Zo)€dz, 
where all the integrals are taken round the small contour. 



Now j / (zo) dz =f(zo) j dz 



=/(zq) X the increase in value of z after once* 
describing the small contour 

= 0. 
Similarly j/(zo) (z - z^) dz = lf(Zo)jd {(z - Zof] = 0, 

when the integral is taken round the small contour. 

Thus, if 7} be the greatest value of | e | for points on the small contour, 
we have 

jf(z)dz^^rfl\z-Zo\\dz\, 

where the integrals are taken round the small contour. 

Now the right-hand side of this equation is clearly of the order rjl^ of small 

quantities. The value of jf(z)dz, taken round the small contour, is there- 
fore a small quantity of order tflK 

Now the number of such small contours ip a given area is of the order 

1 . ' 

T^. If 77' be the maximum value of 7} for all the small contours in the 

area, we see therefore that the total sum of the integrals for all the small 

contours in the area is at most of the order r)'l^ Xj^ or 1;'; and 17' can be 

made indefinitely small by decreasing I. 

It follows, therefore, that the sum of the integrals round all the small 



36] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 49 

contours is zero; that is, the integral round the original contour is zero, 
which establishes Cauchy's result. 

Corollary 1. If there are two paths ZqAZ and ZoBZ from Zq to Z, and if 

f(z) is a regular function of z at all points in the area enclosed by these two 

rz 
paths, then / f(z)dz has the same value whether the path of integration is 

z^AZ or ZqBZ. This follows from the fact that ZqAZEzq is a simple contour, 
and so the integral taken round it (which is the diflference of the integrals 
along z^AZ and z^BZ) is zero. Thus, i{f{z) be an analytic function of z, the 

value of I f(z) dz is to a certain extent independent of the choice of the 

arc AB, and depends only on the terminal points A and B. It must be 
borne in mind that this is only the case when f{z) is an analytical function in 
the sense of § 35. 

Corollary 2. Suppose that two simple closed curves C^ and Ci are given, 
such that Cq completely encloses Cy, as e.g. would be the case if Ci and Ci 
were coucentric circles or confocal ellipses. 

Suppose moreover that f{z) is an analytic function, which is regular at 
all points in the ring-shaped space contained between C^ and (7i. Then by 
drawing a network of intersecting lines in this ring-shaped space, we can 
shew exactly as in the theorem just proved that the integral 



ff(z)dz 



is zero, where the integration is taken romfid the whole boundary of the ring- 
shaped space; this boundary consisting of two curves Co and Ci, the one 
described in a positive (counter-clockwise) direction and the other described in 
a negative {clockwise) direction. 

Corollary S. And in general if any connected region be given in the 
^- plane, bounded by any number of curves C?o, Ci, C„ ..., and if f(z) be 
any function of z which is regular everywhere in this region, then 



j/(z) dz 



is zero, where the integral is taken round the whole boundary of the region; this 
boundary consisting of the curves Cq, Ci, ... , each described in such a sense that 
the region is kept either always on the right or always on the left of a person 
walking in the sense in question round the boundary. 

An extension of Cauchy's theorem I f{z)dz=0, to curves lying on a cone whose vertex 
is at the origin, has been made by Raout {Nouv. Annales de Math. (3) xvi. (1897), 
w A. 4 



50 THE PROCESSES OF ANALYSIS. [CHAP. IIL 

pp. 365-7). Osgood {BvU. Amer, Math, Soc,, 1896) has shewn that the property jf{z) dz—0 

may be taken as the defining-property of an analytic function, the other properties being 
deducible from it. 

Example. A ring-shaped region is bounded by the two circles U| = l and U|=2 in the 

/dz 
— , where the integral is taken round the boundary 

of this r^on, is zero. 

For the boundary consists of the circumference |«| = 1, described in the clockwise 
direction, together with the circumference !«| = 2, described in the counter-clockwise 
direction. Thus if for points on the first circumference we write «=«**, and for points on 
the second circumference we write z=2e^, then B and (j) are real, and the integral becomes 



-«» t.^de . f^ i.2e^d4> 

2e^ 



Jo ^' ^Ji 

or -2frt + 2fri, i.e. zero. 



37. The vaiiie of a function at a pointy expressed as an integral taken 
round a contour enclosing the point. 

Let (7 be a contour within which f(z) is a regular function of z. 

Then if a be any point within the contour, the expression 

z— a 

represents a function of z, which is regular at all points within the contour C 
except the point z^a, where it has a singularity. 

Now with the point z^sa as centre, describe a circle 7 of very small 
radius. Then in the ring-shaped space between 7 and C, the function 

/(f) 

z — a 
is regular, and so by Corollary 2 of the preceding article we have 

f fiz)dz r f(s)dz ^^^ 

J c Z'-a J y z — a 

where I and I denote integrals taken in the positive or counter-clockwise 
sense round the curves C and 7 respectively. 

But (§35) f/(fM^,f/(«)-K^-")/'(«) + '(^-«)rf, 

^ Jy z-a Jy z—a ' 



37, 38] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 51 

where € is a quantity which tends to zero when the radius of the circle 7 is 
indefinitely diminished. Thus 

Jcz-a -^^^lyZ-a Jy Jy 

Now if at points on the circumference 7 we write 

z — a- re^, 
where r is the radius of the circle 7, we have 



f dz P^ir^^de . r^' ,^ ^ . 



and f dz=^jire^d0 = O; 

I f I 
also I edzl^rj. iirr, 

where 17 is the greatest value of |€| for points ^ on 7; and therefore in the 
limit when r is made indefinitely small we have 



/. 



edz = 0. 
y 



"- L4^-2"/<»). 



C 

or 



•^ ^ ^ 27n J c z — VL 

This remarkable result expresses the value of a function /(^) at any point 
a within a contour (7, in terms of an integral which depends only on the value 
of f(z) at points on the contour itself. 

Corollary, If f(z) is a regular function of ^ in a ring-shaped region 
bounded by two curves C and C, and a is a point in the region, then 

-^ ^ "^ zmjoz — a imjcz — a 

where C is the outer of the curves and the integrals are taken in the positive 
or counter-clockwise sense. 

38. The Higher Derivates. 

The quantity /'(-gr), which represents the limiting value of 

f{z + h)-f{z) 

h 

when h tends to zero, is called the derivaie o{ /(z). We shall now shew that 
/' (z) is itself an analytic funcHon of z, and consequently Usdf possesses 
a derivate. 

4—2 



52 THE PROCESSES OF ANALYSIS. [CHAP. HI. 

For if (7 be a contour surrounding the point z, and situated entirely 
within the region in which f{z) is regular, we have 

*=o 27riA \j c Z'-a — h j c z — a \ 

= J^r/M^+LimitA[ /(^) ^^ 

Now f /(^)^^ 

J c(z — ay{z — a-'h) 

is a finite quantity, since the integrand 

(z-ay{z-a-h) 

is finite at all points of the contour (7, and the path of integration is of finite 
length. Hence 

Limit ^-; <^ rT = 0» 

and consequently /' (a) = ^ j ^^^ > 

a formula which expresses the value of the derivate of a function at a point 
as an integral taken round a contour enclosing the point. 

From this formula we have, if h be any small quantity, 

f (a + h) -f (g) 1 [ mdz \ 1 1_) 

h 2m} c h \{z-a-hy {z-af) 

2mJc {z-a-hy(z-ay 

zmj c {z — o) 
where ^ is a quantity which is easily seen to remain finite as h tends to zero. 

Therefore as h tends to zero, the expression 

/'(a+h)-f'ia) 
h 

tends to a limiting value, namely 

2 /■ /{z) dz 
2'iTiJc{z— ay ' 



38] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 63 

The quantity /' (a) is therefore an analytical function of a ; its derivate, 
which is represented by the expression just given, is denoted by /" (a), and is 
called the second derivate of /(a). 

Similarly it can be sheAvn that f" {a) is an analytical function of a, 
possessing a derivate equal to 



3 r f{z) dz ^ 
iJciz-a/' 



this is denoted by /'" (a), and is called the third derivate of /(a). And in 
general an nth derivate of/ (a) exists, expressible by the integral 

n\ f f (z) dz 

and having a derivate of the form 

(n+1)! 



l)!r f{z)dz , 
27rt ic{z-ay^' 

this can be proved by induction in the following way. 

Then 

/(»)(a + A)-/(») (a) n\ [ J(z)dz ( 1 1 ] 

h i-rnio h l(«-a-A)»+' (^-a)*+'j 

^ n!_f f{z)dz U h \-»->_,) 

2iriJc(^-a)"*'AlV e-a) ) 

(n + l)! f f{z)dz 
+ terms which vanish when h tends to zero. 

which establishes the required result. 

A function which possesses a first derivate at all points of a region in the 
^-plane therefore possesses derivates of all orders. 

Exam'ple 1. Verify the theorem 

by use of Taylor's Theorem. 
By Taylor's Theorem we have 



54 THE PROCESSES OF ANALYSIS. [CHAP. III. 



f dz 
But when i: is an integer other than unity, I ^ _ ^ is zero, since 



resumes 



its original value after describing the contour. So the only surviving part of the right- 



hand side is -zr- ./H (a) / , or/(*) (a). 

Example 2. Verify the same theorem by means of integration by parts. 

We have 

nl[ f{z)dz _ ( {n-\)\ f{z) \ (n-iy. f r{z)dz 

f(z) 
and the first term is zero, since 7-- ^. resumes its original value when z makes the circuit 

(z—ap 

of the contour C. Proceeding in this way, we have 

n! f f(z)dz ^ 1 f f!!Hl)dz 

39. Taylor's Theorem. 

CoDsider now a function f{z\ which is regular in the neighbourhood 
of a point z^a. Let C be the circle of largest radius which can be drawn 
with a as centre in the ^-plane, so as not to include any singular point of 
the function f{z)\ so that f{z) is a regular function at all points of G. Let 
5sa + A be any point within the circle G. Then by §37, we have 

^/ IN If f{e)dz 
•' ^ ^ zm Jc z — CL — h 

But at points z on the circle G. the modulus of ^ , will not exceed 
^ z—a—h 

some finite quantity M, 

Therefore • 

1 r /{zydz.h""^^ M.^irR /[A_|\«+i 

27riJa(^-a)«+^(^-a-A) ^ 2ir \RJ ' 

where R is the radius of the circle G, so that 27ri2 is the length of the path 
of integration in the last integral, and R^\z — a\ for points z on the cir- 
cumference of G, 

The right-hand side of the last inequality tends to zero as n increases 
indefinitely. We have therefore 



39] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 56 

which we can write 

« 

This- result is known as Taylor's Theorem] the proof we have given is due 
to Cauchy, and shews exactly for what range of values of z the theorem 
holds true, namely for all points z which are nearer to a than the nearest 
singularity of/ {z). It follows that the radios of convergence of a power-aeriee 
%8 always such as jtist to exclude from, the circle of convergence the nearest 
singularity of the function represented by the series. 

At this stage we may introduce some terms which will be frequently 
used. 

If/(a) = 0, the function /(-^) is said to have a zero at the point -^ = a. 
If at such a point/' (a) is diflferent from zero, the zero of /(a) is said to be 
simple; if, on the other hand, the quantities /'(a), /''(a), .../<**~^* (a) are all 
zero, so that the Taylor's expansion of f(z) at ^ = a begins with a term 
in {z — ay*, then the function f(z) is said to have a zero of the nth order at 
the point z=^a. 

Example 1. Find a functiou /(«), which is regular within the circle C of centre at the 
origin and radius unity, and has the value 

a-oos^ sin^ 

a«-2aco6^+l a*-2acos^+l 
(where a>\ and B is the vectorial angle) at points on the circumference of C, 

We have 

f{z) dz 

c ^•^^ 






« **• . tad . —3 — r :r-— , puttmg «■■«•* 

^j, since the only non-zero term is that from k^n. 



a^ 
Therefore by Maclaurin's Theorem*, 

or /(«)«- • for all points within the circle. 

This example raises the interesting question, What isf(z) for points outside the circle? 
Is it still — ? This will be discussed m §§ 41, 42. 

«0 

Example 2. Prove that the arithmetic mean of all values of «"* I a^«f, for points « 

on the circumference of the circle |£|«1, is a«, if Sa^^*" is regular at all points within the 
circle. 



♦ The result /W=/(0) + */'(0)+^/"(0)+.... 



2 
which is obtained by putting asO in Taylor's Theorem, is osoally called Maclaurin*i Theorem, 



56 THE PROCESSES OF ANALYSIS. [CHAP. III. 

• f{v) (0) 

Let 2 avz^=if{z)y so that a^^ — \- . Then the required mean is 



— • I —^^Tr- J where G is the circle, 



or 

27rt, , 

/(H) (0) 

or J \ I 



ni 
or a,j. 



f » 



Example 3. Prove that if A is a given constant, and (1 - 2zh+h^)~^ is expanded in the 
form 

l+hP^(z)+h^Pi{z)-{-¥P^(z) + (A), 

where F^ {z) is easily seen to be a polynomial of degree n in Zy then this series converges so 
long as ; is in the interior of an ellipse whose foci are the points s=l and z^-l, and 

whose semi-major axis is 5 (A+ j) . 

Let the series be first regarded as a function of h. It is a power-series in A, and 
therefore converges so long as the point h lies within a circle on the A-plane. The centre 
of this circle is the point A=0, and its circumference will be such as to pass through that 
singularity of (1 -2aA+ A*)"* which is nearest to A=0. 

But l-2zh+h^=(h'Z+^^^) (A-«- V^^-^l), 

so the singularities of (1-2M4-A*)"* are the points A=«-(«*-l)* and A««+(«*-l)*, 
at which it is infinite. 

Thus the series (A) converges so long as |A| is less than either 

U-(2«-l)*| or !«+(«»- 1)*|. 

Now draw an ellipse in the z-plane passing through the point z and having its foci at 
the points 1 and - 1. Let a be its semi-major axis, and 6 the eccentric angle of z on it. 

Then z=acoB6+i{a*-'l)^Bin6y 

which gives z±{s^- l)*={a±(a*- 1)*} (cos d+t sin 6\ • 

so |«±(22-l)*|-a±(a8-l)*. 

Thus the series (A) converges so long as A is lees than the least of the quantities 
a'^{a^- 1)* and a-{a^— 1)*, i.e. so long as A is less than a—ia^- 1)*. But 

A=a— (a*-l)* when <*=3(^+i)- 

Therefore the series (A) converges so long as ^ is within an ellipse whose foci are I and 
- 1, and whose semi-major axis is - I^+t ) • 

40. Forma of the remainder in Taylors Series, 

The form found in the last article for the remainder after n terms in 

Taylor's series is 

f(z) JC^dz 






40, 41] THE FUNDAMENTAL PBOPERTIES OF ANALYTIC FUNCmONS. 57 

It is not difficult to derive from this expression the forms of the remainder 
usually given in treatises on the Diflferential and Integral Calculus. For 

on mtegrating by parts the quantity n I ^>— — _ .x^-h , we have 
Jo (z-a- 1)"^^ {z - a)«+^ ^Kn-i-^) ]^ ^^ ^a^ty^^ 



k^ h^ 



1 



"• / - " _ W4.« •"•••> 



{z - a)»*+i {z - a)'»+« 

by successive repetition of this process, 

A« ' 

"" (2r — aY{z -a-hy 

which is a new form for the remainder. 

Now suppose that all the quantities concerned are real. Then along the 
line of integration, {h — ^)**~^ has a fixed sign, so 



^»=(^?i)i/„*<'^-'>"-''^' 



where H lies between the greatest and least values of/<**>(a + ^) between 
< = and t = k We can therefore write H =/<*»> (a + Oh), where < d < 1, 
and then 

or iJ = ^/(»)(a + ^A), 

which is Lagrange's form for the remainder, 

Darboux gave in 1876 {Journal de Math. (3) ii. p. 291) a form for the remainder in 
Taylor's Series, which is applicable to complex variables and resembles the above form 
given by Lagrange for the case of real variable& 

41. The Process of Continuation, 

Near every point P{z^ at which a function f{z) is regular, we have 
seen that there is an expansion for the function as a series of ascending 
positive integral powers of {z — z^, the coefficients in which are the suc- 
cessive derivates of the function at z^. 

Now let A be the singularity o{ f(z) which is nearest to P, Then the 
circle within which thus expansion is valid has P for centre and PA for 
radius. 



58 THE PROCESSES OF ANALYSIS. [CHAP. in. 

Suppose that we are given the values of the function at all points of the 
circumference of this circle, or more strictly speaking, of a circle slightly 
smaller than this and concentric with it : then the preceding theorems enable 
us to find its value at all points within the circle. The question arises, How 
can the values of the function at points ouUide the circle be found ? 

In other words, given a power-series which converges and represents a 
fwnction only at points within a circle, to derive from it the values of the 
function at points outside the circle. 

For this purpose choose any point Pi within the circle, not on the line 
PA, We know the value of the function and all its derivates at P,, from 
the series, and so we can form the Taylor series with Pi as origin, which 
will represent the function for all points within some circle of centre P,. 
Now this circle will extend as far as the singularity which is nearest to Pi, 
which may or not be A ; but in either case, this new circle will generally* lie 
partly outside the old circle of convergence, and for points in the region 
which is included in the new circle hut not in the old circle, the new series tvill 
furnish the values of the function, although the old series failed to do so. 

Similarly we can take any other point P^, in the region for which the 
values of the function are now known, and form the Taylor series with P, 
as origin, which will in general furnish the values of the function for other 
points at which its values were not previously known ; and so on. 

This method is called continuation'^. By means of it, starting from a 
representation of a function by any one power-series we can find any number 
of other power-series, which between them furnish the value of the function 
at all points where it exists ; and the aggregate of all the power-series thus 
obtained constitutes the analytical expression of the function. 



Example, The aeries 



represents the function 



1 £ £« «3 



' ^ ' a-z 



only for points z within the circle \z\=a. 

But any number of other power-series exist, of the type 

1 , r~6 (z-by (z-bf 
a-6"^(a-6)2'*"(a-6)s'*"(a-6)*"*"'*'* 

which represent the function for points outside this circle. 

* The word "gfenerallj" must be taken as referring to the cases which are likely to come 
under the student's notice before he reads the more adyanoed parts of the subject, 
t In Qerman, Fortsetzung. 



42] THB FUNDAMENTAL PROPERTIES OF ANALYTIC FCTNCTIONS. 59 

On functions to which the continuation-process cannot be applied. 

It is not always possible to carry out the process of continuation. Take as an example 
the function f{z) defined by the power-series 

/(«)=1 +««+«♦ -f^^+^W-H.. .+««" + ..., 

which clearly converges in the interior of a circle whose radius is unity and whose centre 
is at the origin. 

Now as t approaches the value +1 by real values, the value of /(«) obviously tends 
towards -foo ; the point +1 is therefore a singularity of f{z). 

But /W=^*+/W, 

80 if « is such that -?*=1, and therefore /(«*) is infinite, then f(z) is also infinite, and so 
« is a singularity off{z) : the point «=■ - 1 is therefore a singularity of /(«). 

Similarly since 

we see that if « is such that z*— 1, then zis & singularity oif{z) ; and in general, any root 
of any of the liquations 

«8=1, 5*-l, «8-l, «i«-l, ..., 

is a singularity of f(z). But these points all lie on the circle |«| = 1 ; and in any arc 
of this circle, however small, there are an infinite number of them. The attempt to 
carry out the process of continuation will therefore be frustrated by the existence of this 
unbroken front of singularities, beyond which it is impossible to pass. 

In such a case the function f(z) does not exist at all for points z situated outside the 
circle \z\»l ; the circle is said to be a limiting circle for the function. 

42. The identity of a function. 

The two series 

1 + ^-1- ^« + ^+... 

and - 1 + (2: - 2) - (-2 - 2)« + (-? - 2)» - (^ - 2)* + . . . 

are not simultaneously convergent for any value of z, and are distinct 
expansions. Nevertheless, we generally say that they represent the same 
functiony on the strength of the fact that they can both be represented by the 

same rational expression . 

This raises the question of the identity of a function. Under what 
circumstances shall we say that two different expansions represent the sa/me 
function ? 

We shall define a function, by means of the last article, as consisting of 
one power-series together with all the other power-series which can be 
derived from it by the process of continuation. Two diflferent analytical 
expressions will therefore be regarded as defining the same function if they 
represent power-series which can be derived from each other by continuation. 

It is important to observe that a single analyticcU eicpression can represent 
different functions in different parts of the plane. This can be seen from the 
following example. 



60 THE PROCESSES OF ANALYSIS. [CHAP. IH. 

Consider the series 

The sum of the first n terms of this series is 



1 / 1\ 1 



The series therefore converges for all finite values of z. But since when 
n is infinitely great, z^ is infinitely small or infinitely great according as | -? | 
is less or greater than unity, we see that the sum to infinity of the series is 

z "when |-2r|<l, and - when 12:|>1. This series therefore represents one 

z 

fwnction at points in tlie interior of the circle | ^r | = 1, and an entirely different 

ftmction at points outside the same circle. 

Example. Shew that the series 

« + g2 + ^+ ,^2+ ... 

, 1 / 2z 2 1 / 2zy .241 / 2g V \ 

2\l-^ 3"3- Vl-«V 3;6-6' Vl-zV -J 

represent the same function in the common part of their domain of convergence. 

43. Laurent's Theorem, 

A very important extension of Taylor s Theorem was published in 1848 
by Laurent ; it relates to the expansion of functions under circumstances in 
which Taylor s Theorem cannot be applied. 

Let C and C" be two concentric circles of centre a, of which C is the inner; 
and let f{z) be a function which is regular at all points in the ring-shaped 
space between C and G\ Let a + A be any point in this ring-shaped space. 
Then we have (§ 37, Corollary) 

y(. + A) l.f -Af) ,, l_.r _/(?L d,, 
•^ ^ ^mjcz — a — h 2'mJcZ^a — h 

where the integrals are supposed taken in the positive or counter-clockwise 
direction round the circles. 

This can be written 



+ 






. 43] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 61 

We find, as in the proof of Taylor's Theorem, that 

[ f{z)dz,h^^' f f(z)dz{z-a)^ 

tend to zero as n increases indefinitely ; and thus we have 

b b 

where a^^^f^I^, and K ^ ^. f J^ - aT'^ (z) dz. 

This result is Laurent* 8 Theorem; changing the notation, it can be 

expressed in the following form : If z be any point in the ring-shaped space 

within which f{z) is regular, and which is bounded by the two concentric 

circles C and C" of centre a, then f (z) can be expanded at the point z in the 

form 

h h 

An important case of Laurent's Theorem arises when there is only one 
singularity within the inner circle C", namely at the centre a. In this case 
the circle C can be taken to be infinitely small, and so Laurent's expansion 
is valid for all points in the interior of the circle C, except the centre a. 

Example 1. Prove that 

1 f^^ 

where •^n(^)=5- I coB{n6^xsmB)d3. 

For Laurent's Theorem gives 

z z 

where 0^=5^. f Z^*"^^ :^, and bn=J-. [ J^"'^ ^'^dz, 

and where C and C" are any circles with the origin as centre. Taking C to be the circle of 
radius unity, and writing z^e , we have 



2trtyo 
= — I cos (nd - ^ sin d) cW, 



62 THE PROCESSES OF ANALYSIS. [CHAP. HI. 

since the parts of / sin (n^-a;sin 6)dB which arise from B and ^ir-6 will destroy each 
other. Thus 

Now ft»= (-!)*«»> since the function expanded is unentered if -- be written for z, 

z 

Thus 

5«=(-i)"y,(x), 

which completes the proof. 

Example 2. Shew that, in the annulus defined by 

\a\<iZi<\bU 

the expression -j , . yr — x[ can be expanded in the form 

where 5.- S^ i^.n-:mi^. '[l) • 

For by Laurent's Theorem if C denote the circle |«|«»r, where |a|<r<|6|, then the 
coefficient of z* in the required expansion is 

Putting z=re^f this becomes 



or 



2.r j, * '^ '^l 2*.*! -p-? 201 H~- 



The only terms which give integrab different from zero are those arising from k^l+n. 
So the coefficient of z^ is 

1 p*^^ 1.3. ..(2^-1) 1.3... (2 ^ +2n-l) <^ 
Sirjo I 2'.^! 2'+-.(";+n)! &'+»' 

Similarly it can be shewn that the coefficient of -- is S^a^, 



Example 3. Shew that 



e'**'*"«-ao+«i«+«8«*+— +- + J+...> 



where *'»*^2ir / ^"■*"*'^*^*<^{(^-*^)8"^^-w^}^> 

and ^»=o^- I ^'''^^'''*^ooa{{v-' u) sine -n6}de. 

^ir J Q 



44] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 63 

44. The nature of the singularities of a one-valued function. 

Consider now a function f{z) which is regular at all points of a certain 
region in the -e-plane, except a point z = a;so that the point a is a singularity 
of the function /{z)* 

Surround the point a by a small circle 7, with a as centre. Then in the 
ring-shaped space between 7 and some larger concentric circle (7, the function 
/ (z) can by Laurent's Theorem be expanded in the form 

^0 + -4 1 (^r — a) + ^a (-2^ - a)* + -4 , (^ — a)* + . . . 

z --a {z — ay (z — ay 

The terms in the last line are called the Principal Part of the expansion of 
the function at the singularity a ; if they were non-existent, the function 
would clearly be regular at the point ; so they may be regarded as consti- 
tuting the analytical expression of the singularity. 

Now these terms of the Principal Part may be unlimited in number, 

i.e. the series 

Si B^ Bi 

z — a {Z'-ay {z — ay 

may be an infinite series ; in this case the point a is said to be an essential 
singularity^ of the function /(z). Or on the other hand, they may be 
limited in number, i.e. the series just written down may be a terminating 
series ; so that the expansion can be written in the form 

Bn Bn-i _^ ^_Bi,^A, + Ai{z''a)'\'A2{z^ay+.... 



{z-aY {z-af^^ '" z-a 

In this case the function is said to have a pole of order n at thelpoint a. 
When n is unity, so that the expansion is of the form 

— ^H-ilo-hili(-?-a)-hil,(-^-a)"+..., 
z — a 

the singularity is said to be a simple pole. 
Example 1. Find the singularities of the function 

c 



z 



Near 2=0, the function can be expanded in the form 

e CM afi 



^ a a^ a* 



* The name essential singularity is also applied to any singularity of a one-valued ftmotion 
which is not a pole, i.e. to singularities for which no Laurent expansion at all can be found. 



64 THE PROCESSES OF ANALYSIS. [CHAP. III. 



c 

e~ 
or 



— ^ e « ( - + c, ) +positive powers of z. 



There is therefore a simple pole at 2=0. Similarly there is a simple pole at each 
of the points ^trnia {n= +1, +2, +3, ...). 

Near 2=0, the function can be expanded in the form 

c 
gt-a 



B-a 



14.-1-+ - + 

or ^ 



e 



(>^"-?--)-' ' 



which gives an expansion involving all positive and negative powers of {z - a). So there is 
an essential singularity at 2= a. 

There is also an essential singularity at 2=00 , as will be seen after the explanations of 
the next article. 

Example 2. Shew that the function defined by the series 

, >^-'{(i+y-i ' 

has a simple pole at each of the points 

/ 1\ ^"^ 
z^il + -\e * (it«0, 1,2, ...7i-l; ?i=l, 2, ...00). 

(Cambridge Mathematical Tripos, Part II., 1899.) 

45. The point at infinity. 

The behaviour of a function f(z) for infinite values of the variable £ 
can be brought into consideration in the same way as its behaviour for finite 
values of z. 

For write ^ = -> , so that the infinite values of z are represented by the 

point / = in the /-plane. Let f(z)^<f> (z). Then the function ^ (/) may 
have a zero of order m at the point 2:' = ; in this case the Taylor expansion 
of ^ (/) will be of the form 

and so the expansion oi f{z) valid near z^(x> will be of the fonn 






T" • • • • 



In this case, / (z) is said to have a zero of order m at z==oo , 



45] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 65 

Again, the function ^ (/) may have a pole of order m at the point / = ; 
in this case, 

and so for large values of z,f{z) can be expanded in the form 

In this case, ^ = oo is said to be a pole of order m for the function /{jb). 

Similarly f(z) is said to have an essential singularity at -^ = oo , if ^ {£) 

has an essential singularity at the point z = 0. Thus the fiinction eF has an 

1 
essential singularity at 5 » oo , since the function ^ or 

has an essential singularity at / = 0. 

Example, Discuss the function represented by the series 

The function represented by this series has singularities at z^--^ and f~ — -y 

(n«l, 2, 3, ...), since at eacb of these points the denominator of one of the terms in the 
series is zero. These singularities are on the imaginary axis, and are infinitely numerous 
near the origin ^=0 : so no Taylor or Laurent expansion can be formed for the function 
valid ia the region immediately surrounding the origin. 

For values of z other than these singularities, the series converges absolutely, since the 

ratio of the (n+l)th term to the nth is ultimately . > ^ , which is veiy small when n 

is larga The function is an even function of z (i.e. is unchanged if the sign of ^ be 
<:hanged}, is zero for all infinite values of 2, and m regular at all points outside a circle 
C of radius unity and centre at the origin. So for points outside this circle it can be 
expanded in the form 

-where, by Laurent's Theorem, 

** 2ni J a n=ow^ «"■** + «* 

0-a«^tt-l ^»-3^-2«/ a-2n ^-4n ^-e» \ 

^^"^ n\(a'^+z^)" ^1 V""i5~ "*""?"" 15-+-J' 

1 (-!)*-! a"** 
and the coefficient of - on the right-hand side of this equation is ^ — -. . 

Z 7Cr • 

W. A. 5 



66 THE PROCESSES OF ANALYSIS. [CHAP. lU. 

Therefore, since only terms in - can fiimish non-zero integrals, we have 



*ifc=o— • ^ I — i 

^ 2mn^oJc n\ z 



j_ 

=(-l)*-ie«". 

Therefore for large values of z (and indeed for all points z outside the circle of radius 
unity) the function can be expanded in the form 

JL J. J_ 

The function has a zero of the second order at z== ao , since the expansion begins with 

a term in -5 . 
z^ 

46. Many-valued functions. 

In all our previous work we have supposed the function f(z) to have one 
definite value corresponding to each value of z. 

But functions exist which have more than one value corresponding to 

each value of z. Thus the function ^ has two values (viz. + VJ and — V^) 
corresponding to each value of z, and the function tan~* z has an infinite 
number of values, expressed by the formula tan~^ z ± hrr, where k is any 
integer. 

We may however for many purposes consider + Vz and — VJ as if they 
were two distinct functions, and apply to either of them separately the 
theorems which have been investigated in this chapter. When we in this 
way select some one determination of a many- valued function for considera- 
tion, it is called a branch of the mimy-valued function. Thus the values 
log z, log z + 27rt, log z + 4iTn, . . . , would be said to belong to different branches 
of the function log z. 

There will be certain points for which the values of the function given by 
diflferent branches coincide: these points are called branch-points of the 
function, and must be included among its singularities. Thus the function 
jgr* has a branch-point at jgr = 0, since either branch there gives the same value, 
zero, for the function. 

It must not however be supposed that the branches of a many-valued 
function really are distinct functions. The following example shews how 
the different branches of a many-valued function change into each other. 

Let f{z) = A 



46] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 67 



Write 2:=r(co8^ + i8ind), where 0<^<27r. Then the two values of 
f{z) are 

e . . e\ , ,-/ . . . 0' 



+ Vr [cos ^ + i sin 5] and - Vr [cos ^ + i sin 5 j . 



Let us take the former of these values, and consider its changes as the 
point z describes a circle round the origin {z = 0). As the point travels, r is 
unchanged, but constantly increases, and when the point reaches again the 
starting-point after completing the circuit, has increased by 27r. The 
function has therefore become 

r { + 27r . . ^ + 27r\ 
-h Vr (cos — 2 — + ^ sin — — j , 

^r — Vr (cos^ + isin ^j. 

In other words, the branch of the function with which we started has passed 
over into the other branch. 

In following the succession of values of f{z) along a given path, the final 
value is deduced without ambiguity from the initial value; and all con- 
ceivable paths lead to one of two final values, viz. 'Jz and — Vz, But it 
appears from the above that it is not possible to keep these branches per- 
manently apart as distinct functions, because paths lead from one value to 
the other. 

The idea of the different branches of a function helps us to understand many of the 
''paradoxes'' of mathematics, such as the following. 

Ck)nsider the function 

du 
for which t- =^ (1 +log «). 

When z is negative and real, -^ is not real. Now if ^ is a negative quantity of the 
form \^ (where p and q are positive or negative integers), u is real. 
If therefore we draw the real curve 

we have for negative values of « a series of conjugate points, arranged at infinitely small 

intervals of z : and thus we may think of proceeding to form the tangent as the limit of 

du 
the chord, just as if the curve were continuous ; and thus -r- , when derived from the 

inclination of the tangent to the axis of a?, would appear to be real The question thus 

dtt 
arises, Why does the ordiniiry process of differentiation give a non-real value for -r- ? The 

5—2 



68 THE PROCESSES ^OF ANALYSIS. [CHAP. IH. 

explanation is, that these conjugate points do not all arise from the same branch of the 
function u=^. We have in fact 

and log z has an arbitrary additive p£urt 2lrirt, where k is any integer. To each value of k 
corresponds one branch of the function u. Now in order to get a real value of u when z is 
negative, we have to choose a suitable value for k : and this value ofk varie$ atvfego from 
one conjugate point to an adjacent one. So the conjugate points do not represent values of 
u arising from the same branch of the function ubs^*, and consequently we cannot expect 

diL 

the value of -r- to be given by the tangent of the inclination to the axis of x of the 
tangent-line to the series of conjugate points. 

Example 1. If log z be defined by the equation 

log i?a Limit n{f^—\\ 



»=• 



shew that log 2 is a many- valued function, which increases by 2n-t when z describes a 
closed path round the origin. 

For put «=r (cos 6-\-imi 6). 

Then one of the vcJues of log 2, on this definition, is 

Limit n if* (cos — Hisin- |-lL 
»=■« I \ n M/ , J ' 

1 
where r* is the positive nth root of r. 

This can be written 

1 
Limit n {r* - 1} + i6, 

naoo 

« 

When z describes a closed path round the origin, the first term in this expression 
remains unaltered, while the second increases by 2frt ; hence the result. 

Example 2. Find the points at which the following functions are not regular. 

{a) «*. Answer, «— 00. 

(6) cosecs. Answer, z=0, ±ir, ±2ir, ±3ir, .... 

z-\ 



^^^ «»-5«+6- 


Answer, z^2,Z, 


1 
{d) e\ 


Answer, «=0. 


(6) {(^-1)4*. 


Answer, «=0, I, 00. 



Example 3. Prove that if the different values of a», corresponding to a given value of z 
are represented on an Argand diagram, the representative points will be the vertices of an 
equiangular polygon inscribed in an equiangular spiral, the angle of the spiral being 
independent of a. 

(Cambridge Mathematical Tripos, Part I., 1899.) 



47, 48] THE FUNDAMENTAL PEOPERTIES OF ANALYTIC FUNCTIONS. 69 

47. LtouvUle's Theorem. 

We know by § 38 that if /(^) be any function of z which is regular at 



all points of the ir-plane within a circle G, of centre a and radius r, then 

^ ^^^ 27^tjc(^-a)«+^• 



Now let M be the greatest value of \f(z)\ at points on the circle 0. 
Then this equation gives (§ 32) 

nlM 

From this inequality an important consequence can be deduced. Suppose 
that /(z) is, if possible, a regular function of z over the whole z-plane, 
including infinity, ie. that it has no singularities at all. 

Then in the above equation M is finite when r is infinite, whatever n is ; 
and therefore /<*^ (a) is zero for all values of n and a, i.e. /(a) is a constant 
independent of a. We thus arrive at Liouville*8 theorem^ that the only 
fwnction which is regular everywhere is a constant. 

As will be seen in the next article, and again frequently in the latter half of this 
yolnme, Liouville's theorem furnishes short and convenient proofis for some of the most 
important results in Analysis. 

48. Functions with no essential singularities. 

We shall now shew that ^ ofnly one-valued fu/nctions which have no 
singularities in either the finite or infinite part of the plane, except poles, are 
roMonal functions. 

For let / (z) be such a function ; let its singularities in the finite part 
of the plane be at the points Ci, Cj, ... c*: and let the principal part (§44) 
of its expansion at the pole Cy be 



Z-'Cr (z^CrY (^-Cr)"^ 

Let the principal part of its expansion at the pole £: = oo be 

if 5 = 00 is not a pole, but a regular point for the function, then the coefficients 
in this expansion will be zero. 

Now the function 

^^^ r%\z-Cr^iz-Cry^-^{z-Crrr] 



70 THE PROCESSES OF ANALYSIS. [CHAP. III. 

has clearly no singularities at the points Oi* ^> ••• Cjb, x ; it has therefore no 
singularities at all, and so by Liouville's theorem is a constant ; that is, 

f{z) = constant 4- Ojir + (V* + ... + (ti^ 
f{z) is therefore a ratioual function, and the theorem is established. 






Miscellaneous Examples. 

1. Obtain the expanBion 

2. Obtain the expansion 

/(.)-/W+"i^°[/-W+/'(.)+!{/'(.+^)+/'(»+^)+- 

+ .... (Corey.) 

3. Obtain the expansion 

+ ... (Corey.) 

4. In order that values U-\- Vt, which are given as continuous functions of the arc 
of a circle, should be the boimdary values of an analytic function, shew that it is necessary 
and sufficient : 

(a) That — '^ — ^^^ — — — -^ at the place ^=0 should be uniformly integrable for 

all values of a ; 
{h) That the values of 7 shall be given by 

V{a)^^ i' {U{a'-^)'U{a^^)}Qot^d^. (Tauber.) 



MISC. EXS.] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 71 
5. Shew that for the series 



«i»0 



2»+«-*' 



the region of convergence consists of two distinct areas, namely outside and inside a circle 

of radios unity, and that in each of these the series represents one function and represents 

it completely. 

(Weierstrass.) 

6. Shew that 

(Jaoobi & Scheibner.) 

7. Shew that 

+*"<"*+^^-;-^'"+**>(i-»)-'/'V(i-<)«'-t dt. 

(Jacobi & Scheibner.) 
'8. Shew that 

(i-^-.;^.,.-«-.*.=i;{,.=±|.*....<:-:|tg*-;)^-.} 

^c--'-' ,i:t>y.y/s.':'j :'--<'-'-'-''"- 

(Jacobi & Scheibner.) 

9. I( in the expansion of (a+Oi^+tv')*^ by the multinomial theorem, the remainder 
after n terms be denoted by i^ so that 

(a+ai«+a^«)«»=ilo+i4i2+ii22*+...+-4n-i«*"^+^ 
shew that 

i2»(a+ai*+<V«)-j^ (^^^^^^pri <«• 

(Jacobi & Scheibner.) 

10. If (ao+ai«+a^)-*-* j{ao+a^t+a^t*)'^dt 
be expanded in ascending powers of z in the form 

shew that the remainder after (n— 1) terms is 

(Jacobi & Scheibner.) 

11. Shew that the series 

2 {l + X,(.)e'}-^jr-, 



nao 



where X,»(«)= -l+^-gj + ^j -...±^, 



72 THE PROCESSES OF ANALYSIS. [CHAP. III. 

and where 4> (^) is a regular function of z near «aO, is convergent in the neighbourhood of 
the point «=0; and shew that if the sum of the series be denoted bj /{$), then /(f) 
satisfies the differential equation 

/' {e) -/(«) - (z). (Pincherie.) 

12. Shew that the arithmetic mean of the squares of the moduli of all the values of 
the series 2 Oj^^ on ^ circle |f|»r, situated within its circle of convergence, is equal 



to the sum of the squares of the moduli of the separate terms. 

(Qutzmer.) 

13. Shew that the series 

00 

converges when |f | < 1 ; and that the function which it represents can also be represented 
when \z\ < 1 by the integral 

/a\4 /■• $'» dx 

\w/ Jo' ^-z X*' 

and that it has no singularities except at the point f ~ 1. (Lerch.) 

14. Shew that the series 

2 2 ( z «-i 1 

- (* + ^*) + - 2 l^j _ 2^_ 2^,^.^ (2v+ 2v'«)« ■*■ (1 - 2^- 2^z-H') (2p+2p'z'hyi ' 

in which the summation extends over all integral values of p, i/, except the combination 
(iraaO, V »0), convergcs absolutely for all values of z except purely imaginary values ; and 
that its sum is '+ 1 or - 1, according as^the real part of z is positive or negative. 

(Weierstrass.) 

15. Shew that sin •Ite ( « + -H can be expanded in a series of the type 

Z Zr 

in which the coefficient of either t^ or z'^ia 



5-1 sin (2tt00s ^) cos nBdB, 



16. If 

hew that /(f) is finite and continuous for all real values of f, but cannot be expanded as 
Maclaurin's series in ascending powers of z ; and explain this apparent anomaly. 



CHAPTER IV. 
Thb Uniform Convergence op Infinite Series. 

49. Uniform Convergence. 

We have seen* that the sum of a convergent series of analytic functions 
of a variable z can have discontinuities as z variea It was found by Stokes "f 
and Seidell in 1848 that this can never happen except in association with 
another phenomenon, that of non-uniform convergence, which will now be 
investigated. 

Consider the series 

« ?! . ^ + ^(2^-1) 

( 1 + 2^) ( 1 + 2^ + ^2) "^ ( 1 + 2z + ^) ( 1 + 3^ + ^) "^ • • • 

. 5 + g^U-l) 

We shall first shew that this series is convergent for all values of z except 
certain isolated points. 

For, except for the roots of 1 + w-r + ^ = 0, the nth term can be put in 

the form 

1 1 

l+n^ + -8^ l+(n + l)^H- sf^-^^ ' 
so the sum of the first n terms is 

o 1 1 

**- 1 + 2z i + (n + 1)« + -2*+* ' 

which, as n becomes infinitely great, tends to the value ^ ^ for all points 
except 5 = 0: and for -r = 0, we have /S = 0. 

Thus (except at the roots of the equations I + nr + ^ = 0) the series 
converges ; and it represents a regular function, except at ^ = 0, where it has 
a discontinuity. 

* In § 42. 

t Collected Paper$, Vol. i. p. 286. 

t Miineh. Ahh. 



74 THE PROCESSES OF ANALYSIS. [CHAP. IV. 

What lies at the root of the discontinuity ? 
The remainder after n terms is 

For ordinary values of z, say ^ = 1, this remainder decreases rapidly as 
n increases. Thus if w = 10, j? = 3, the remainder = o j. . on > * negligible 
quantity. But now let z approach near to its discontinuity 0: say 
^~ 1000000* "r^®^ ^i^'^ ^^^ value of z, the remainder after 1000 terms is 
nearly 1, and the remainder after 1000000 terms is still nearly s- This 

shews that, as z approaches the discontinuity at ^=»0, the terms which 
contribute sensibly to the sum tend to recede to the infinitely distant part o/ the 
series, so the first 1000 terms do not furnish a good approximation at all. 

We can express this analytically as follows : — The number of terms n, 
which we have to take in order to make |i2n| less than a given small positive 
quantity e, tends to oo as we approach the point of discontinuity. 

This circumstance is the basis of the following definition : — 

Let Ui{z)-\- u^{z)'\-Ut{z)-\-u^{z)'\- ,.. 

be a series of functions of z, which is convergent at all points z within a given 
area in the j?-plane. Let Rn be the remainder after n terms. Then since 
the series converges, if we take a small finite quantity e we can find at any 
point on the area a number r (varying from point to point) such that \Rn\ < € 
so long as n > r. If the numbers r corresponding to the aggregate of points 
in the vicinity of a given point z are all less than some definite finite number, 
the series is said to be uniformly convergent at the point z ; but if near any 
point z the number r tends to infinity, so that no definite upper limit can be 
assigned to it, the convergence of the series is said to be non-uniform* in 
the neighbourhood of the point z. 

Example 1. Shew that the series 

■*■ 1 +a* "^ (1 +««)«'^ •'• ■*"(TT?)* "*"'"• * 

which converges absolutely for all real values of is, is discontinuous at z—O and is non- 
uniformly oonvei*gent in the neighbourhood of ^— 0. 

The svun of the first n terms is easily seen to be 1 +«*— ,-^ — sxr;:^ . So when z is not 
zero the siun is 1 +^, and when z is zero the sum is zero. 

* An interesting geometrical treatment of uniform convergence is given by Osgood in VoL lu. 
of the BvU. of the Amer, Math, Soe, p. 59 (1896). 



49] THE UNIFORM CONVEROENCE OF INFINITE SERIES. 75 

The remainder after n terms is 7rTT2\»=i* "^^^ ^^^^ ^ made smaller than any 

log- 
aasigned smaU finite positive quantity c by choosing n so that n-l> j j m . But as 

9 tends to aero, . 7|^r^\ tends to infinity, so n must tend to infinity, i.e. we have to 

include an infinite number of terms in order to get the remainder less than c . This series 
is therefore non-uniformly convergent in the neighbourhood of e=0. 

Example 2. Shew that at «=0 the sum of the series 

z z z 



l(z+l) ' (a+l)(25+l)"-^{(n-l)«+l}{fw+l} 
is discontinuous and the series is non-uniformly conveigent. 

The sum of the first n terms is easily seen to be 1 ^ : so when z is zero the 

sum is 0. 

The remainder after n terms of the series is — — : ; so when z is nearly zero, say 

fs one-hundred-millionth, the remainder after a million terms is --^^ or 1 - =^ , so 

100^ 
the first million terms of the series do not contribute one per cent of the sum. And in 
general if i be small, it is necessary to take n large compared with the large quantity 

- in order to make the remainder after n terms smalL There is therefore non-uniform 

z 

convergence in the neighbourhood of 2=0. 
Example 3. Discuss the series 

»=i{l+»M{l-Kn+iy«««}- 

The nth term can be written ; 5-^ - _— ^ Tr^— , , so the sum to infinity is r— — 5 , 

l-»-nV l-|-(n-|-l)'«* l-l-«* 

and the remainder after n terms is z— ^ tth-, . 

1-H(n-|-1)*2* 

However great n may be, i^we take z equal to -tt> ^^^ remainder will have a finite 

value, namely \ ; the series is therefore non-uniformly convergent at 2^0. 

NoTB. In this example the sum of the series is not discontinuous at ««>0. 

Cayley* regards non-uniform convei*gence as consisting essentially in the occurrence of 
a discontinuity in the sum of a series. The condition for a discontinuity in a series 

at the point z^av& that the series 

T^\ a-z 
shall have an indefinitely large sum when (a— z) is indefinitely small. 

* ** Note on Uniform Oonvergeaoe," Proc, Hoy, 80c. Edinb. xix. (1891-2), pp. 208-8. 



76 THE PROCESSES OF ANALYSia [CHAP. lY. 

Thus in the series 

(l-^)+«(l-;?)+z«(l-«)+..., 

which is non-uniformly convergent and discontinuous at ^a*!, we have 



a — z 



= -«*, when a=l, 



80 the sum of the series 2 ^^-^ is ; — , which is infinite for «= 1. 

60. Connexion of discontinuity with non-uniform convergence. 

We shall now shew that ths sum of a series of continuous functions of z, 
if it is a uniformly convergent series for values of z within certain limits, 
cannot he discontinuous for values of z within those limits. 

For let the series be /(2r) = 1^1 (^) + 1^3 (<2:)+...+ 14^ (2r)+... = flfn(^) + lJnW> 
where B^ is the remaiDder after n terms. 

Since the series is uniformly convergent, we can to any small positive 
number e find a corresponding integer n independent of z, such that 

I iZn (-3^) I < « for all values of z within the area^. 

Now n and € being thus fixed, we can, on account of the continuity of 
Sn (z), find a positive number 17 such that, when |5 — / 1 < 17, the inequality 

is satisfied. 

We have then 

<\Sn(z)''Sn(z')\ + \Rn(z)\'^\Itn{z')\ 

which establishes the fiesult. 

Exam^ 1. Shew that at «=0 the series 

1 1_ 

where Wj («)==«, «»(«)=«*•"*-«*•"•', 

and real values of z are concerned, is discontinuous and non-uniformly convergent 

1 
The sum of the first n terms is i^~^ ; as n tends to infinity, this quantity tends to 
1, 0, or - 1, according as 2 is positive, zero, or negativa The series is therefore absolutely 
convergent for all values of z, and has a discontinuity at f =0. 

1 
The remainder after n terms, when z is small and positive, is 1 -i^'^ ; however great 

n may be, by taking «=c~(*»~i) we can cause this remainder to take the valuie 1 - - , which 

is different from zero. The series is therefore non-uniformly convergent at «>bO. 



50, 51] THE UNIFORM CONVERGENCE OF INFINITE SERIES. 77 

ExamjiU 2. Shew that at 2=0 the aeries 

n-i{H-(l+^)*-i}{l +(!+«)*} 
is discontinuous and non-uniformly conveigent. 

The nth term can be written 

l-(l+g)* _ l-.(l+g)*-i 
l+("l+2)"* l+(l+z)»-i' 

the sum of the first n terms is n . /i , w • Thus considering real values of t greater 

than - 1, it is seen that the stun to infinity is 1, 0, or - 1, according as 2 is negative and 
greater than -2, zero, or positive. There is thus a discontinuity at 2«>>0. This discon- 
tinuity is explained by the fact that the series is non-uniformly convergent at «=0 ; for 
the remainder after n terms in the series when t is positive is 

-2 

and however great n may be, by taking e=- this can be made to take the value 

— 2 

, which is difilerent from zero. The series is therefore non-uniformly convergent 



so 



1+e 
at «=0. 

61. Distinction between absolute and uniform convergence. 
The uniform convergence of a series does not necessitate its absolute 
convergence, nor conversely. Thus the series (§ 49, Ex. 1) S .^ ^.^ con- 
verges absolutely, but (at ^ = 0) not uniformly : while if we take the series 

« (, l)n-l 

its series of moduli is 



which is divergent, so the series is only semi-convergent ; but for all real 
values of 2, the terms of the series are alternately positive and negative and 
numerically decreasing, so the sum of the series lies between the sum of its 
first n terms and of its first (n + 1) terms, and so the remainder after n terms 
is less than the nth term. Thus we only need take a finite number of terms 
in order to ensure that for all real values of z the remainder is less than any 
assigned quantity, i.e. the series is uniformly convergent. 

Absolutely convergent series behave like series with a finite number of 
terms in that we can multiply them together and transpose their terms. 

Uniformly convergent series behave like series with a finite number of 
terms in that they are continuous and (as we shall see) can be integrated 
term by term. 



78 THE PROCESSES OF ANALYSIS. [CHAP. IV. 

62. Condition /or uniform convergence, 

A sufficient though not necessary condition for the uniform convergence 
of a series may be enunciated as follows : — 

If for all values of z within a certain region the moduli of the terms of a 
series 

8 = v^{z) -k- v^{z) '\- u^ W + ••' 

are respectively less than the corresponding terms in a convergent series of 
positive constants 

then the series 8 is uniformly convergent in this region. This follows from 
the fact that, the series T being convergent, it is always possible to choose n 
so that the remainder after the first n terms of T, and therefore of 8, is less 
than an assigned positive quantity € ; and since the value of n thus found 
is independent of z, the series 8 is uniformly convergent. 

GoroUary. The theorem is still true if the moduli of the terms of <S, 
instead of being less than the terms of T, are to them in a variable but finite 
ratio. 

Example. The series 

COSZ+gjCOS*;? + ^COS'« +... 

is uniformly convergent for all real values of z, because the moduli of its terms are not 
greater than the corresponding terms of the convergent series 

1+^+1+ 

whose terms are positive constants. 

63. Inteffration of infinite series. 

We shall now shew that if 8(z)=^iii(z) + ti^{z)-\' ... is a uniformly con- 
vergent series of continuous functions of z, for values of z contained within 
some domain, then the series 



jtifi(z)dz'\-jv^(z)dz + ... , 



where all the integrals are taken along with some path C in the domain, is 
convergent, and has for sum l8(z) dz. 

For let n be some definite finite number, and write 

8 (z)=^ U,(Z) + U^(z)-¥ ... '\' Un(z) + Rn{z), 



so 



j 8(z)dz== jtii(z)dz+...'^ Iun(z)dz + lRn(z)dz. 



52, 53] THE UNIFORM CONVEBGENCE OF INFINITE SERIES. 79 

Now since the series is uniformly convergent, to every positive number € 
there corresponds a number r independent of z, such that when n^ r we have 
R^{z)\< €, for all values of z in the area considered. 

Therefore if 2 be the length of the path of integration, we have (§ 32) 



/' 



<d. 



Rn{z)dz 

Therefore the modulus of the difference between I S(z)dz and the sum 

of the n first terms of the series X I Un (z) dz is less than any positive 
number provided n is large enough. This proves both that the series 
2 jt^(z)dz is convergent, and that its sum is J8(z)dz, 

Example 1. As an example of the necessity of this theorem, consider the series 

• 2z{n(n + l)sin*«2- l}co8«' 

2 



««i {l+n''sin2«8}{l + (n+l)«8in««2}* 

The nth term is 

2zn cos 2* 2z(n+l) cos z* 

l+n«sin2 2«"' ! + (« + !)* sin^z*' 
and the sum of n terms is therefore 

2gcosg' 2g(n+l)cosg* 

The series is therefore absolutely convergent for all real values of ;; : but the remainder 

after n terms is 

2z{n+l)co8z'^ 

H.(n+l)2sin2?' 

and if n be any number however infinitely great, by taking z= — -i this has the finite value 
2. The series is therefore non-imiformly convergent at ;?«0. 

Q* COS Z^ 

Now the sum to infinity of the series is i— .—5-,, and so the integral from to « of 

the sum of the series is tan~^ (sin ^). On the other hand, the sum of the integrals from 
to i? of the first n terms of the series is 



tan~* (sin a^) - tan** (n-fl sin «*), 

and foms= 00 this tends to 

tan~*(sin«')— ^. 

Therefore the integral of the sum of the series differs frx>m the sum of the integrals of 
the terms by •= . 

Example 2. Discuss the series 



• 2g^{l-n(e-l)+e*'^^^} 
,2in(n+l)(l+^«^)(l+e»+»««) 



for real values of z. 



80 THE PROCESSES OF ANALYSIS. [CHAP. IV. 

The nth term of the series may be written 



n(l+e^«) (n+l)(l +«*+»««)• 
The sum of the first n terms is 

1+^" (71+ 1)(1 +€•*!««)• 

The series therefore converges to the value ^ ; and since the terms are real and 

ultimately of the same sign, the convergence is absolute. The integral from to « of the 
sum of the series is 

log (1 +«?«). 

The sum of the first n terms of the series formed by integrating the terms of the series 
is 

log(l+e0«)-^log(l+e»*ii^), 
which fornsoo tends to 

l0g(l+6««)-l 

This discrepancy is accounted for by the non-uniform convergence of the series at f «0 ; 
for the remainder after n terms in the original series is 

or 



(n+l)(l+«-^) «_+?,-t+(n+i); 



i 

and however great n may be, on taking «=> -— -r this takes the value unity ; so the series 
is non-uniformly convergent at f ^0. 

Example 3. Discuss the series 

W1+W8+W3+..., 

where 

Uj ■= ze-'^f tt» = nw-"** - (n - 1) -w^*-*)**, 

for real values of z. 

The sum of the first n terms is nzer*^, so the sum to infinity is for all real values 
of z. Since the terms Un are real and ultimately all of the same sign, the conveigenoe 
is absolute. 



In the series 



/ riidz+l t^dz+ I i(^dz-\',„f 
Jo Jo Jo 



the sum of n terms is ^ (1 - 0"**^), and this tends to the limit ^ as n tends to infinity ; this 
is not equal to the integral from to i? of the sum of the series 2 u^. 

The explanation of this discrepancy is to be found in the non-uniformity of the 
convergence neariS^O, for the remainder after n terms in the series Wj+Wj-I-... is -Twe-*^ ; 

and however great n may be, by taking 2= - we can cause this to tend to the limit - 1, 

which is different from zero: the series is therefore non-imiformly convergent near z=0. 



54, 55] THE UNIFORM CONVERGENCE OF INFINITE SERIES. 81 

64. Differentiatimi of infinite series. 

The converse of the last theorem may be thus stated : 

If 8 (z) ^ Ui(z) -h u^{z) + , .. is a convergent series of analytic functions of z, 
which are regular when the variation of z is restricted to be within a certain 

domainyandiftheseries'S.{z)^-r'V^{z)-\--j-ii>2(z) + ... is wniforrrdy convergent 
within this domain, then this latter series represents -j- 8 (z). 

For by the preceding result, if a and z are two points within the domain, 
we have 

r X(t)dt = r ui if) dt + \\i (t)dt+,.. 

J a J a J a 

Since 

if^(z) + u^(z)+ ,,. and i^i(a) + ti,(a) + ... 

are each of them convergent series, we can write this 



J c 



^{t)dt = {i^i (z) + iLi{z) + .,.]-'{ui (a) + Wa(a) + ...} 



= 8(z)^8(a), 
and hence \ 

We may note that a derived series may be non-imiformly convergent even when the 
original series is uniformly convergent : for instance the series 

sin«— J8in2«+Jsin3«+.., 

is non-uniformly convergent at z=^Tr; although the series from which it can be derived, 
namely 

- cos a+5jCOS 22- ^ cos 3«+ ..., 
is uniformly convergent for all real values of z. 

66. Uniform convergence of Power-8eries, 

We shall now shew that a power-series is wniformly convergent at all 
points within its circle of convergence. 

For let jR be a region, forming part of the area of the circle, and let r be 
a quantity greater than the modulus of every point of i2, but less than the 
radius of convergence. Then if z be any point of R, the moduli of the terms 
of the series 

ao + aiZ + a^-^- ... 
w. A. 6 



82 THE PROCESSES OF ANALYSIS. [CHAP. IV. 

are less than the moduli of the corresponding terms of the convergent series 

ao + ctir + a,i'*+ ... . 

But the latter series does not involve z, and so (§ 52) the power-series is 
uniformly convergent within the region R\ as R is arbitrary, the series there- 
fore converges uniformly at all points within the circle of convergence. 

It must be observed that nothing is proved regarding points on the 
circumference ; we do not even know that the series is convergent there at all. 

Corollary, A power-series is continuous within its circle of convergence : 
and the series obtained by differentiating and integrating it term by term 
are equal to the derivate and integral of the function respectively. 

Example, As an example of this, consider the series 

which is convergent at all points within a circle of radius 1. We can integrate it term by 
term, so long as the path of integration lies in this circle ; the result is 



/; 






Now / T-T-h clearly represents that value of tan~i» which lies between - ^ *^d +^ , 
So the series represents this value of tan" ^« and no other. 



Miscellaneous Examples. 

1. Shew that the series 






represents . .^ when \z\<l and represents >, when |«|>1. 

Is this fetct connected with the theory of uniform convergence ? 

2. Shew that the series 

2sin- +4sin;r +...+2*sin ;;— + ... 

converges absolutely for all values of z, but does not convei^ uniformly near «bO. 

00 

3. If aserie8^(«)«r j (c^- c,,+j) sin (2i» + !)««• (in which Cq is zero) converges uni- 

IT 

formly in an interval, shew that g{z) -. is the derivate of the series 

/(«)= 2 — sin2i«ir. (Lerch.) 



CHAPTER V. 
The Theory of Residues : Application to the 

> 

Evaluation of Keal Definite Integrals. 

66. Residties. 

If a point ^ = a is a pole of order m for a function /(z), we know by 
Laurent's theorem that the expansion of the function near 2r = a is of the form 

where ^ (s) is regular in the vicinity of ^ = a. 

The coeflScient a»i in this expansion is called the residue of the function 
f{z) relative to the pole a. 

Consider now the value of the integral \f{z) dz, where the integration is 

taken round a circle 7, whose centre is the point a and whose radius is a small 
quantity p. 

We have I f(z)dz=l, a^r f 7 ;z + / <l>{z) dz. 

Now I <l> {z) rf^ = 0, since <f>(z)iB a regular function in the interior of the 
circle 7 : and (putting z — a^ pe^) we have 

, = p-»^-i r^ , when r + 1 

= 0, when r^\. 
But when r=! I we have 

f J^ = ride = 2^. 

jyZ — a Jo 

Hence finally / f(z) dz = 2'rna^i. 

Jy 

Now let C be any contour, containing in the region interior to it a number 

6—2 



84 THE PROCESSES OF ANALYSIS. [CHAP. V. 

of poles a, 6, c, ... of a function /(^), with residues a_i, 6_i, c_i, ... respec- 
tively : and suppose that the function f{z) is regular at all points in the 
interior of (7, except these poles. 

Surround the points a, 6, c, ... by small circles a, /8, 7, ... : then since 
the function f(z) is regular in the region bounded by (7, a, ^8, 7, ..., its 
integral taken round the boundary of this region is zero. But this boundary 
consists of the contour (7, described in the positive sense, and the contours 
•1 A 7> ••• described in the negative sense. 

Hence = [ f{z) dz -| f{z) dz-^j f{z) dz..,, 

or ^ ~ I f{^)^^ ■" 27rwi«i — 27rt6_i ••• • 

J c 

Thus we have the theorem of residues, namely 

f{z)dz^'l7ntR, 



L 



C 

where 2-B denotes the sum of the residues of the function /{z) relative to 
those of its poles which are situated within the coo tour 0. 

This is an extension of the theorem of Chapter III. § 36. 

67. Evaluation of real definite integrals. 

A large number of real definite integrals can be evaluated by the use of 
contour-integrals and the theorem of residues. The following examples will 
serve to illustrate the various ways in which these aids to the evaluation 
may be applied. 

Example 1. To find the values of 

P%«**co8(n^-sm^)cW and (^ e"^^ sin {n3- am 6) dS, 
Denoting these integrals respectively by / and •/, we have 









Write 0^=2, and let (7 be a circle of radius unity round the origin in the f -plana Then 
«8 6 assumes the sequence of real values from to 2n-, 2; describes the circle C, 

Hence l-iJ^-. I eFz'^'-^dz 

^ J c 

=s 27r X the residue of -—ry at «=0 

_2.r 



1 



57] THE THEORY OF RESIDUES. 85 

Therefore -^■" -r i 

Example 2. The method used in Example 1 can be very generally applied to 
trigonometrical integrals taken between the limits and 2v, As another example, 
consider the int^pral 

/-r f-, (a>6). 

Write efi^^z'y and let C be the circle on the z-plane whose centre is at the origin and 
whose radius is unity. 

/ [ 2cb 



Then 



dz 



» Jc 
=4ir X simi of residues of j-^ — r. at poles contained within C. 



Now 



fea»+2€W+6 2^a^-b*- 



-1 



a V a* — 6* a v a* — 6* 



Thtt^ore the two poles are at 2= - ^ r^- and f ■■ - ^ + ^ ^ ~ , and the residue 

at the former (which is the only one within C) is -- — — . 

Hence /« 



Example 3. Shew that 

(a+6cos5)2~(a2-'62)>/2' 
Example 4. Find the value of 



i: 



/ 



* x sin ma? , 
ax. 



-00 ^+a* 

Let (7 be a contour formed by the real ckxis together with a semicircle y, consisting of 
that half of a circle, whose centre is at the origin and whose radius is very large, which 
lies above the real axis. 



Then ^ - j is a function of z which has only one pole in the interior of C, namely at 

,-— , dz = 2*rt X residue of -, ^ at its pole at. But writing 

z^at-^-^y we have 

^— ma 

=-o-v + positive powers of f. 



86 THE PROCESSES OF ANALYSIS. [CHAP. V. 

Therefore ^ ^ = -y^—-ir+ positive powers of («-ai). 
Thus the residue of -5- — = at at is - «"*•. 



The«fo« .ir~-/^ -;^,«fc=(/^^ +/J /g,^ 



Since 






is infinitesimal compared with pr at points on y, the integral round y is 



infinitesimal compared with / — 

Therefore wte"* 

Equating imaginary parts, we have 



or 2fr, and is therefore zero. 



/* iT sin mx , _^ 

Example 6. To find the value of 

/ ««~"**sin(asin6^)^— ^. 

Take a contour C composed as in Example 4 of an infinite semicircle y and the real 
axis. 

Then / — .«<" ^ . ^ cfe—2»ri x residue of 77-. «** « . • at its poles inside C 

But r\««« , A has only one pole in the interior of C, namely at the point z^rL 
Now if 2«an+f, we have 

2i^ ?+7i = 2i'^ 2nf+T* = 4^'^ + positive powers off. 

1 6r 

Therefore the residue is t-^' ' 

But at points on y, e^«*=0, so e<*« =1, and so 

Jy2i a^+f^ 2tjy^"2* 

Therefore ^eo^-^^^^./" e^ooBhx^j^^amibx) ^^, 

or JVoo.ta8m(a8in6*)J;^=^ (««-»'-!). 

We may note that in the above 1 stands for the limit of 1 where k is infinitely 

/■* 
great, and is not equal to the limit of 1 where k and I are different. 



EXS.] THE THEORY OF RESIDUES. 87 

Example 6. Prove by integrating 



/: 



^dz 



round the contour used in Examples 4 and 5, that 
Example 7^ 'Find the value of 



/, 



* mnmx . 



Consider a contour C, formed of 1<> a semicircle r whose centre is at the origin and whose 
radius is very large, 2^ a semicircle y whose centre is also at the origin and whose radius is 
very small, and 3** the portions of the real axis intercepted between these circles. The semi« 
circles are to be drawn in the upper half of the «-plane, i.e. the half above the real axis. 



. , . — 5rs«2frt X the residue of -7-,-:— -v at the singularity 



Zmeat. 



But if we write 2= at + (> where C is small, we have 



Thus the residue at at is - 



Therefore - 



4a» 



(-!)• 



ia? 



(**+a)' jc «(«»+««)« ~U-. "^jr JvJ*(»«+a«)»- 



Now / a . jNg is infinitely small at points on r, so the integral taken round r vanishes. 

_co«(2:«+a*)* a* 20* \ a/ 
In this, I means f + f > where the two c's are the same : but in the final result 

J -00 J 9 J —00 

we can put f >bO, since the final integrand is finite at the origin. 

Equating the imaginary parts on both sides of this equation, we obtain 

/* sin mxdx w irg"*^ / 2\ 
-ooi"(^+a«)«"^" 2a» [^'^aj' 

J /"• Binmxdx it ire"^^/ . 2\ 

And so I —7-5 oTo^TT-i 7— r— |w*+-|. 

Jo^{x^-^ay 2a* 4a» \ a/ 

Example 8. Find the value of 



/:. 



* cos 2ax - cos 26^ , 

1^ '^- 



Take the contour C formed as in Example 7 by an infinite semicircle r, a small semi- 



88 THE PROCESSES OF ANALYSIS. [CHAP. V. 

circle y round the origin, and the parts of the real axis intercepted between them. Within 
this contour the function — |- has no singularities. 

In this equation i must be r^arded as an abbreviation for i + i where f is the 
radius of y. 

^8«<* 1 

Now at points on r, —^ is ssero compared with - , so the integral round V is zero. 



z* " z 



-— cb a one-half of 2tnx the residue of — ,- at the origin 



am X the residue of = 



= — 2ira* 

—^ dz=-27ray 

— 00 ^ 

^ cfe=2fr(6-a). 

Taking the real part of this we have 

/* cos 2a:r - cos 26^ j ^ ,, \ 
^ flte=2»r(6-a), 

, . cos 2cu? - cos 2&r . i, .^ i ^ j i _j__- x T* ^ 

and smoe -^ is nmte when ^=0, we need no longer restnct / to mean 

Example 9. Find the value of 

/o'--«^(T-^)^f^ («>0). 

We have J^ ^-i sin (^ - 6*) ^^ 

Consider a contour (7, formed as in Examples 7 and 8 by an infinite semicircle r, 
a small semicircle y round the origin, and the parts of the real axis intercepted between 
them. 

Then i j (-«y»-ic««^^=2irtxthe residue of | (-^O^'^^^^T^ft^^ i^ singularity 



«=ir. 



EXS.] THE THEORY OF RESIDUES. 89 

Putting 2arn+C <^^ n^ecting powers of (y we see that the expansion of 



begins with a term 



"4— f— ' 



80 the required residue is — j r*-* c-^. 

Therefore s^"*«"^=5 / (-^t)"-^ 

z z J c 



rdz 



At points on r the integrand is infinitesimal compared with - , and so the integral 

z 

round r is zero. 

At points on y the integrand is approximately ^—^ — z^^ and so if a > the integral 
round y is zero. 






Therefore / ^^s- (^-fe-)^-g, = | /^J-^T-.- 

/* /it: 

gaooBtegin (a sin 6^) — . 
^ 

Wehave f Voo«to8in(asin6x)~ =i f * «<»^— , 

Jo 'a; 2i j_ao 4? ' 

where in the latter int^ral i must be regarded as an abbreviation for i +1 where 

€ is a small quantity. 

Take a contour C, consisting as in Examples 7, 8, 9, of an infinite semicirole r, a small 
semicircle y of radius t roimd the origin, and the parts of the real axis intercepted between 
them. 

Then 0= [ e^^^ [ eo.^^- ( ea^^+ T e^"^ . 

J C X JT ^ Jy ^ J '• ^ 

At points on r, we have 6*«*=0, tf<»<^=l, and so 

J T X JTX 

At points on y, 6***= I, so 

Jy ^ Jy^ 

Therefore [" ga«fc»<?ff =^(c«-i), 

/oe J 

^oo«6xgin(asin&ar) - =3(e«-l). 

round the same contour as that used in Examples 

7, 8, 9, 10, shew that ('^?^^cLv=Z. 

jo ^ 2 



90 THE PROCESSES OF ANALYSIS. [CHAP. V. 

Example 12. To find the value of 

I :r-— cKr, and/ , dx (0<a<l). 

Jo 1+^ jo 1-^ 

Write /=/ f— dar,andfi'-/ f—dx. 

Jo l+a? J Q I -X 

Ab will be seen from the working below, the integral K has a meaning only when I is 

+ I , where / is a small positive quantity, 
i-f^ Jo 

Consider a contour C formed of (a) that half r of a circle, whose centre is at the origin 
and whose radius is a large quantity 72, which is above the real axis, (b) that half y of a 
circle whose centre is at the origin and whose radius is a small quantity r, which is above 
the real axis, (c) that half y' of a circle, whose centre is at the point ( - 1) and whone 
radius is a small quantity r', which is above the real axis, (d) the parts of the real axis 
intercepted between these semicircles. 

— , where the many- valued function is supposed to have that one of 

its determinations which is real and positive when z is real and positive. The integrand is 
regular in the interior of the contour (7, and so 

f 2f-^dz 
Now on y the integrand is sensibly equal to 2i^'\ and so the integral to — L which 

-r 

is infinitesimal, since a > 0. 

(-1)0-1 

On y, the integrand is sensibly equal to ^ J^ — ; putting 1 + « «■ r'tf*', the integral 

1 "rZ 

along y' is / %d$^ or iV ( - 1)«~*. 

On r, the integrand is sensibly equal to -^^ , the modulus of which is infinitesimal 

m 

compared with , -: ; so the integral along F is zero. 



Thus iri=( -!)*-« /+Jr=-/(cosair-tsincwr) + ir. 

Therefore equating real and imaginary parts, we have 



sin air ' 



K= IT cot air. 

Example 13. By using the result 

f'^ x'-^dx _ 
Jo l+x ~sL 



sm atr 



shew that r-^==^—. Limit 2 -r — . (Kronecker.) 



68] THE THEORY OF RESIDUES. 91 

68. Evaluation of the definite integral of a rational function. 

The principles which have been applied in the preceding paragraph can 
also be used to evaluate an integral of the form 

f f{!c)dx, 

J —00 

where /(a?) is a rational function of x, in the cases when this integral has a 
meaning. 

a ioD\ 
For suppose that f{a) is brought to the form of a quotient Yj-i , where 

g (x) and h (x) are polynomials in x. In order that the integral may have a 
meaning unconditionally, it is necessary that the degree of g (x) should be 
at least two units lower than that of h (x), and that the equation h(x) = 
should have no real roots. 

Consider now a contour C, formed of the real axis together with a 
semicircle F of large radius, whose centre is at the origin, and which lies in 
the upper half of the z-flsLne. 

We have 1 f{z)dz — 2iri x sum of residues oi f{z) at the poles oi f{z) 

J c 

contained within C, 

Now / = I + / • *^^ since f{z) has a zero of at least order 2 at 
jt = 00 , it follows that / is zero. 

Hence I f{x)dx^ 27n x sum of residues of f{x) at those of its poles 

/ —00 

which are contained in the upper half of the ^-plane. 

If the d^;ree of g (x) is lower than that of A (x) by only one degree, or if A (x) has real non- 
repeated roots, the integral will still have a meaning provided we make certain restrictions, 

Le. that I shall be imderstood to mean the limit, when k tends to oo and t to zero^ of 
where c is a typical root of the equation A (:7) =0. 



J'-^L' 



Example I. The function TyTTSs ^^^ ^ single pole in the upper half of the 2-plane, 

3i 
namely at 2»t, and the residue there is -tt.', we have therefore 



/: 



dx 3ir 



(^+1)3 8 * 



Example 2. Shew that / 7 — — , jr-. = 5-1 • 



92 THE PROCESSES OF ANALYSIS. [CHAP. V. 

69. GaiLchys integrals 

We shall next discuss a class of contour-integrals which are very fre- 
quently found useful in analytical investigations. 

Let (7 be a contour in the ^-plane, and let f{z) be a function regular 
everywhere in the interior of C, Let <^ {z) be another function, which in the 
interior of C has no singularities except poles ; let the zeros of <^ {z) in the 
interior of C be Oi, Oj, ..., and let their degrees of multiplicity be rj, r„ ...; 
and let its poles in the interior of (7 be 6i, 6j, ..., and let their degrees of 
multiplicity be «i, 5a, .... 

Then by the fundamental theorem on residues, we have 

^ — ; I f(z) %-T-l dz = sum of residues of -4^-}—- in the interior of C. 
2'inJc <t>{^) <f>(^) 

Now !T^ c*^ bave singularities only at the poles and zeros of 

<f> (z). At one of the zeros, say Oi, we have 

<f>{z) = A(z- OiY' +B(z-aiy^+' + .... 

Therefore <^' (z) = Ar^ (z - o,)'"*-^ + B(n + l){Z''a,y^-{- ..., 
and /(z) =/(a,) + (^ - Oi)/' (aj) + . . . . 

Therefore !f^ = + a' constant + positive powers of (z — Oi). 

<P yZ) Z — Oi 

Thus the residue of !T} > at the point ^ = Oi , is r^f{a^. 
Similarly the residue at -er = 6i is — «i/(6i) ; for near «r =s 6^, we have 

and f{z) =/{bO + (-? - 6,)/' (bi) + • • ^ 

f(z)6' (z) -5,/(6i) . . r r 

so "^ ^ , ; . a= I + a constant + positive powers of jp — 6i. 

<f>{z) z-bi ^ ^ 

' Hence ^ f^/{z) *'|^J dz^t rj(a,) - 2 .,/(60, 

the summations being extended over all the zeros and poles of <f> (z), 

60. The number of roots of an equation contained within a contour. 

The result of the preceding paragraph can be at once applied to find the 
number of roots of an equation <f>{z) = contained within a contour C, 

For on putting f{z) = 1 in the preceding result, we obtain the result that 

^T— ^ I ^"7 \ dz is equal to the excess of the number of zeros over the number 
2mJc<f>{^) 



59 — 61] THE THEORY OF RESIDUES. 93 

of poles of if>(z) contained in the interior of (7, each pole and zero being 
reckoned according to its degree of multiplicity. 

Example 1. Shew that a polynomial (t) of degree m has m roots. 
Let <^(«) = ao2^+ai«^"* + ...+am- 

Then ^ = ^og;r^^ >"-+^- ^ . 



"" _ * y2' " " 



For large values of «, this can be expanded in the form 

*'W = ^ + ^ 
<l>{z) z^z^ 

Thus if C be a large circle whose centre is at the origin, we have 
Hence as ^ (z) has no poles in the interior of C, we have 



number of zeros of d> (z) = ^j— : j %-rl cfe" 



m. 



Example 2. If at all points of a contour C the inequality 

is satisfied, then the contour contains k roots of the equation 

For write /(«)=a^a!~+a„»--i«^~^+...+ai«+aQ. 
Then /(«)=«».» A^ a^^-»+...+«.,, j^+ a^.^-'+...+a, ^ 

=a*«* (1 + i7) say, where | CT"! < 1 on the contour. 
Therefore the niunber of roots of f{z) contained in C 

2irt ; c /W 

"2irijcV« 1 + C^ rfW 
— =27rt ; and since |^| < 1 we can expand (1+ (7)~* in the form 

Therefore the number of roots contained in C is equal to k. 

61. Connexion between the zeros of a function and the zeros ofUe derivate, 

Macdonald* has shewn that if f(z) be a regular function of z in the interior of a 
contour C, defined by an equation \f{z)\=M where M is a constant, then the number of zeros 
of f(z) in this region exceeds the number of zeros of the derived function f'{z) in the same 
region by unity, 

* Proc. Land, Math. Soc, xzix. (1898). 



94 THE PROCESSES OF ANALYSIS. [CHAP. V, 

For since f{i) has no essential singularity in the region, the number N of its zeros in 
the region is finite. Now if m be a small number, the part of the locus |/(«)|=m in the 
interior of the contour C consists of N closed curves surrounding the N zeros of f{z). As 
m increases, these ovals increase, until two of them coalesce, the point at which they 
coalesce being a node on the curve corresponding to that particular value of m. When 
m has increased to its final value if, the N closed curves have coalesced into one closed 
curve, and therefore N^-l nodes have been passed through. Each of these nodes is 
a zero of f (z); for if /(«)=0+«V^, where and yfr are functions of x and y with real 

coefficients, then ^ and -^ vanish at a node on the curve ^+^= constant; that is, 

f'{z) vanishes. Moreover, two ovals cannot coalesce at more than one point, b» f{z) is 
single-valued. 

Hence the number of zeros of /' (z) inside the contour is (^- 1). 

The proof assumes the zeros of f(z) in the interior of (7 to be all simple : the case where 
f{z) has multiple zeros can be at once reduced to this, by dividing out the factor common 
to f{z) €Uid /' (z). If /' (z) has two zeros equal, two of the double points coalesce, that is, 
three ovals coalesce at the same point. 

Similarly it can be shewn that the number of zeros of /' {z) in the region between the 
contours |/(«)|«=mi and |/(«)|"««ij is equal to the number of zeros of f(z) in the same 
region, if /(«) is regular in the region. 

Example 1. Deduce from Macdonald's result the theorem that a polynomial of degree 
n has n zeros. 

Example 2. Deduce from Macdonold's result that if a ftmction /(z), regular for real 
finite values of z, has all its coefficients real, and all its zeros real and different, then 
between two consecutive zeros of f(z) there is one zero and one only of f'{z). 



Miscellaneous Examples. 

1. a function 0(«) is zero for «=0 and regular when I^Kl. If /(x, y) is the 
coefficient of » in <^ (^+yO> prove that 

'o i^2a>>cos^-h ^ -^(^^> sm^)cW-W>W. 

(Trinity College Examination, 1898.) 

« «i. XI- X r* sinew; , 1 e«+l 1 /t j v 

2. Shew that j^ -__<ir-j^y-y--. (Legendre.) 

3. By integrating I e-^dz round the perimeter of a rectangle of which one side is the 
real axis and another side is parallel to the real axis and at a distance a from it, shew that 



/:. 



e-'*cos ^aidt^sfne-^\ 



'00 

and I e-^ Bin 2(Udt^0, 



/: 



oi- XL X [ l-rco8 2^ , . ^j^ w, \-r 

4. Shew that / , — ^^-r^ log sm 6dB = t log —j- 

jo l-2rcos2^+r* ^ 4 ® 4 



MISC. EXS.] THE THEORY OF RESIDUES. 95 

6. Shew that 



/, 



a?cwr=--log(l+a) if -.l<a<l 



l-2aco6a?+a* 4a 

and = ^ log ^1 + ^^ if a« > 1. (Cauchy.) 

6. Shew that 



/, 



00 



sin 01^ sin (bqX sin d>-^ sin o^ , ir . . 



• •• 



if a be different from zero and 



(Stermer.) 



7. If a point z describes a circle C of centre a, any one- valued function u^f{z) will 

describe a closed curve y in the t«-plane. Shew that if to each element of y be attributed 

a mass proportional to the corresponding element of (7, the centre of gravity of y is the 

f{z) 
point r, where r is the sum of the residues of -^^-^ at poles in the interior of C. 



z — a 

(Amigues.) 



8. Shew that 

dx 7r(2q-hft) 



i-«(^ 



-00 (^+ 6^) (^ + a2)« 2a26 {a-^hf ' 
9. Shew that 

dx _ TT 1 .3. ..(2^-3) 1 



/ 



»oo(a+6j?2)* 2*-»6* 1.2...(n-l) a^-h' 

10. If /VW=(l-^)(l-47«)...(l-a;*-i)...(l-ar«)(l--^)... (1-;f«*-%. 

... (l-a?»-i)(l-:c«»-«)... (l-x(»»-»>"), 
shew that the series 

convergee when »■ is not a root of one of the equations 

1-0; 



©■- 



and that the sum of the residues of f{x) contained in the ring-shaped space included 
between two circles whose centres are at the origin, one having a small radius and the 
other having a radius between n and n-|-l) is equal to the number of prime numbers less 
thann-hl. 

(Laurent.) 



CHAPTER VL 
The Expansion of Functions in Infinite Series. 

62. Darbov^'s formula. 

Darboux has given* a formula from which a large number of expansions 
in infinite series can be derived 

Let f{z) be an analytic function of z, regulai* at all points z within a 
circle of centre a and radius r; and let z he a, point within this circle. 
Let (z) be any polynomial in z, of degree w. Then if R^ denotes the 
expression 

(- 1)« {z - a)*»+' C <f> (0/<"-^*' {a + t{z^ a)] dt, 

Jo 

where the integration is taken along the real axis of t, we have on integration 
by parts 

R^ = [\- ir{z- aY <f> {t)f^^ {a + e (^ - a)}J 

+ (- 1)«-^ (z - a)« f'f (O/^*** {a + t(z-- a)} dt, 

J Q 

or i2n = (- lY (^ - ar {<t> (!)/<"» (z) - <t> (0)/<«» (a)} 

+ (- D^^i {z - aY fV' (0/*"^ [a + t(z^ a)} dt 

Jo 

Integrating the last integral by parts in the same way, we obtain 

i2„= (- ir (z - ar {<!> (l)/<~> W - <t> (0)/<«) (a)} 

+ (- 1)~-» (^ - a)*^^ {f (I)/<'^»> (-?) - f (0)/(«-« (a)) + ... 
- (^ - a) (<^<«-^) (1) r (^) - <^'^^^ (0)/' (a)} 

+ (-e - a) f <^ w (0/' [a+tiz"- a)] dt. 
Jo 

Now <^<*> (t) is a constant independent of t, since <^ (t) is a polynomial of 
order n ; and hence 

(z - a) f ' <^^«) (0/' {a + < (^ - a)} dt = <^» (0) {/(z) -/(a)}. 

♦ Liouville'8 Journal (3), u. (1876), p. 271. 



62, 68] THE EXPANSION OP FUNCTIONS IN INFINITE SERIES. 97 

Thus finally we have Darboux's formula 
<f>^^ (0) {f(z) -/(a)} ^(z^a) {<^*-« (1)/' (z) - <^^-« (0)/' (a)]... 

+ (- 1)» (iT - a)^' r (f> (t)/^"^'^ {a + t(z- a)} dt 

Jo 

Taylor's expansion may be derived from this formula by putting 
^(^)aB(t — 1)", and then making n tend to infinity: other new expansions 
may be obtained by substituting special polynomials of degree n for <f> (t), and 
in the resulting formula making n tend to infinity : in each case it must 
of course be shewn that Rn tends to zero as n tends to infinity. 

Example, By subetituting 2n for n in Darbouz's formula, and taking it>{t)=:t^(t- 1)% 
obtain the expansion 

/(«)-/(«)- J^ ^"^^ynr''^V <"W+(-l)'"-V<"(a)}, 

and find the expreesion for the remaioder after n terms in this series. 

63. The BemoulUan numbers and the Bemotdlian polynomials. 

• z z 

If the constants which occur in the expansion of ^ cot ^ fin ascending 

powers of jt be denoted by J3i, jBs, J3t» ... > so that 

z z s^ * s^ ^ 

then Bn is called the nth BemoulUan number. It is found that 

-''1 — g I -^^a — 80 » •'^t = 42 > • • • • 

The BemoulUan numbers can be expressed as definite integrals in the 
following way. 



TTT 1 r* sin ©a? da? S f* . , 

We have I J; , = 2 / €r^^ sm pxdx 



00 



= 2 — *^ — 



= -1+1 






Equating coefficients of j)*'*~' on the two sides of this equation, and writing 
a = 2t, we obtain 

A proof of this result, depending on contour integration, is given by Carda, MoncUthefte 
fwr Math, und Phys. v. (1894), pp. 321-4 

W. A. ' 7 



98 THE PROCESSES OF ANALYSIS. [CHAP. VI. 

BxampU. Shew that 

* ir**(2»»-l)yo "smhF* 

The Bemoullian polynomial of order n is defined to be the coefficient of 

t* e** — 1 . 

—. in the expansion of t -^rZTT ^ ascending powers of t It is denoted by 

<f>n (z), 80 that 

eV-T= 2 ^V (!)• 

6*— 1 »■! n! ^ -^ 

This function possesses several important properties. Writing (z + 1) 
for z in the preceding equation and taking the difiference of the two results, 
we have 

n-1 Tli 

On equating coefficients of t^ on both sides of this equation we obtain 

nz'^^ = <f>n(z+l)-<f>n(z\ 
which is a difference-equation satisfied by the function ^^ (z). 

The explicit expression of the Bemoullian polynomials can be obtained 
as follows. We have 



and 



e*- 




t 


tef + 1 t 


«*- 


l~2c«-l 2 




~2 £ 4 2 




= 2i~*2i-2 



"^"2"^ 21 ~ 4! ■^•••• 



Hence 



„r,~in r"^ 2r"^"3r'*'-|f "2■'"■2^"^■^••r 
From this, by equating coefficients of <*, we have 

the last term being that in ^ or ^ ; this is the explicit expression of the nth 
Bemoullian polynomial. 



mm 



64] THE EXPANSION OF FUNCTIONS IN 'INFINITE SERIES. 99 

The BemouUian numbers and polTnomials were introduced into analysis by Jacob 
Bernoulli in 1713. 

Example. Shew that 

*nW=(-ir*i»(i-«). 

64. The Maclaurin-Bernoullian expansion. 

In Darboux's formula write <^ {t) = ^n (0> where 4>n (0 '^ ^^^ ^^^^ Bemoul- 
lian polynomial 

Now from the equation 

<^(^+l)-.<^„(0 = n^-S 

we have by differentiating k times 

<^n«*»(e + l)-i^n^(0 = w(n-l)...(n-A)r-*-^ 

Putting ^ s in this, we have 

But the value of <^n^' (0) is obtained by comparing the expansion 

<^(^) = 0n(O) + ^<^,'(O)+|j<^"(O) + ... 

with the expansion 

Substituting the values of <^„**(1) and ^n**(0) thus obtained in Darboux's 
result, we find what is known as the Ma^^laurin-Bernoullianformtda, 

(z - a)/' (a) ^f{z) -/{a) - —^ {/' (,z) -/' (a)} 

+ ^i%P^{/"(^)-/"(a)l + -. 

•^ ^" ^^'(In"- 2)7 "^ t/*^ (^) -/"^ («)} 

In certain cases the last term tends to zero as n tends to infinity, and we 
can thus derive an infinite series from the formula. 

Example, lff{z) be an odd function of «, shew that 



JV*.(<)/«***"(-«+2«)<ft, 



2nl 
where ^. (t) is the Bemoullian polynomial of order n. 

7—2 



/ 



100 THE PROCESSES OF ANALYSIS. [CHAP. VI. 

66. Burmann's theorem. 

We shall next consider a number of theorems which have for their object 
ths expansion of one function in powers of another function. 

Let <f> (z) be a function of z, which takes the value b when z takes the 

value a, so that 

b = <t> (a). 

Suppose that ^ (z) is an analytic function of z^ regular in the neighbour- 
hood of the value -e? = a, and that (f/ (a) is not zero. Then Taylor's theorem 
furnishes the expansion 

<l>iz)-b = 4>'(a)(z-a) + ^^iz-ay+..., 
and on reversing this series we obtain 

which expresses -^ as a regular function of the variable {<f>{z) — 6}, for values 
of z in the neighbourhood of a. If then f(z) be a regular function of ^ in 
the neighbourhood of a, it follows therefore that /(-«?) is a regular function of 
{<l> (z) — 6} in this neighbourhood, and so an expansion of the form 

A') =/(«) + «! {* (^) - 6} + Jj {4> (^) - b}* 

will exist, which, as it is a power-series in {^ (z) — 6}, will be valid so long as 

\^(z)-b\<r, 
where r is some constant. 

The actual expansion is given by the following theorem, which is 
generally known as Burmann's theorem. 

If yfr (z) be a function of z defined by the equation 

, . z — a 

then the function f(z) can for a certain domain of valines of z be ea^pa/nded in 
the form 

fiz) =/(a) + 1^ ^^^^i,"^^" ^. [/' (a) tt («)}-] ; 
and the remainder after n terms in the series is 



i_ c' f r <i,(z)-b -\'-^f'(t )<f>'(z)dtdz 



2',rijjyl4>{t)-bj if>(t)-<l>(z) 
where y is a simple contour in the t-plane, enclosing the point < = a. 



66] THE EXPANSION OP FUNCTIONS IN INFINITE SERIES. 101 

To prove this, we have 

- A. ['[ /' (0 <t>' (^) dt dz r ^(z)-h ^ 

iinJJy <f>{t)-b I TW^ 

{^(^)-fel-' -l 

But —['[ \^(_')-i'\ V(t)<l>'(')cUdz {»(^)-6}*-» f /'(t)dt 
ZmJaJyl'l>(t)-bj (f>{t)-b 2in{k+l) Jy{<t>it)-b}'^' 

2^- (A + 1) J^ (< _ a)*+' - 2,ri(A+l)! ^* ^-^ («> t^ ^">5 J' 

Therefore /(^)=/(a)/T^ ^^ii^* £.[/'(«) ftW] 






Exampli 



Lf'f lt(ft:lT~'/l(tl£Mdtdz 

'^JaJyl<l>{t)-bj <f>{t)-<f>(2) • 



where 



27rt 









To obtain this expansion, write 
in the above expression of Burmann's theore^n ; we thus have 



Zi 



But 






(&J '"^•'"•^}« -{^1 ''''^"'^),,, (P^**i^« *=«+') 



=(n-l) ! X coefficient of ^-i in the expansion of tf-«^(>«+<) 

=(n-I)!xooefficientof^-iin i (-l)^^^<^(g«+<r 

r-o rl 

^ ^ A(n-l-r)!(2r-n+l)r 

The highest value of r which gives a term in the summation is r=n- 1. Arranging 
therefore the summation in descending indices r, beginning with r«n- 1, we have 

=(-i)«-ic;. 

which gives the required result 



«•« 



ip 



102 



THE PROCESSES OF ANALTSia 



[chap. VI. 



Excmiple 2. Obtain the expression 



««-sin««+- . 2®^ '+3~5 • 3 8m««+ .... 

Example 3. Let a line p be drawn through the origin in the ;s-plane, perpendicular to 
the line which joins the origin to any point a. It zh% any point on the «-plane which is 
on the same side of the line p as the point a is, shew that 



00 1 /«—a\ '"*■•' ^ 



66. Teixeira's extended form of Bunnann's theorem. 

In the last paragraph we have not investigated closely the conditions of 
convergence of Burmann's series, for the reason that the theorem itself will 
next be stated in a much more general form, which bears the same relation 
to the theorem just given that Laurent's theorem bears to Taylors series: 
viz., in the last paragraph we were concerned only with the expansion of a 
function in positive powers of another function, whereas we shall now discuss 
the expansion of a function in positive and negative powers of the second 
function. 

The general statement of the theorem is due to Teixeira*, whose exposi- 
tion we shall follow in the next two paragraphs. 

Suppose (1) that/(^) is a regular function of ^ in a ring-shaped region Ay 
boundid by an outer curve S and an inner curve s; (2) that 0{z) ia a 
regular function everywhere inside /S, and has a single zero a within this 
contour; (3) that x is the affix of some point within A; (4) that for all 
points of the contour S we have 

l^(a;)|< 1^(^)1, 
and for all points of the contour s we have 

i^(a;)i>i^(^)i. 

The equation 

has, in this case, a single root z=»x in. the interior of S, as is seen from the 
equation 

1 r e'(z)dz _ I [[ ff{z) , ^ ., . r ff{z) i 

^ J_ r ff(z)dz 
^27riJs 0{z) ' 

of which the left-hand and right-hand members represent respectively the 
number of roots of the equation considered and that of the roots of the 
equation 6{z)rsQ contained within 8. 

* CreUe'9 Journal, cxxn. (1900), pp. 97—123. 



mmm 



66, 67] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 103 

Cauchy's theorem therefore gives 

/M = J_ [( A^W('')ds [ Az)0'{z)dz -\ 
^^*^ 2%'7rlJae{z)-e(x) J, '^(z)-0(x)]• 
The integrals in this formula can, as in Laurent's theorem, be expanded 
in powers of 0(ai), by the formulae 

f f{z)ff{z)dz _lff,,.t Az)ff{z)dz 

We thus have the formula 

where a ^ JL { fMli^l^ 

Sn=i^\nz)e^^{z)ff{z)dz. 

This gives a development oi f{x) according to positive and negative 
powers of 6 {x\ valid for all points x within the ring-shaped space A. 

67. Evaluation of the coefficients. 

If the function /(^) has no singularities but poles in the region limited 
by the curve a, the integrals which occur in the preceding formula can 
be evaluated in the following way. 

Let 6i, &9, ... &A be the poles; and let Ci, C2, ... c*, c, be circles with 
centres &i, 6s, ... 6;^, a, respectively, and with very small radiL 

Then A - ^ ( A^)^(')d z_ 1 f /'(z)dz 

"'iiriJs 0^' {z) ~ liri } s ne^ (,zj 

= S JL/" f'i^ldz 1 (f'{z)dz 
and Bn'^'-^j f{z)e^'{z)ff(z)dz 



a?— a 



Ix^a 



104 THE PROCBSSBS OF ANALTS1& [CHAP. VI. 

Thus if Om be the degree of multiplicity of the pole 6,», and if ^ ^ be 
denoted by 0i (x), we have 

" n!L<iB»-'K»(«)jJ, 

+ I 1 r<^- y'(a>)(a>-6m)-^M '| 

It may happen that a is also a pole of /(x). It is easily seen that in 
this case A^, is given by the formula 



■^ (n + /9) ! n [dafi+^ \ ^,» (a;] 



where /3 is the degree of multiplicity of the pole a ; the formula for Bn must 
likewise be replaced by 

- ^inoni [^ ^f <*> ^'" <-) <- - '^>'"-"j].^' 

when w ^ )8. 

The preceding formulae do not give the value of Aq ; this can be found 
from the formula 

A ^ i l_f f{^)^i^)dz . 1 { f{z)ff{z)dz 
"^'"jLiiiTrJc^ 0iz) ^iiirj, 0(z) ' 

which gives 

when a is a regular point {oTf{x) ; and 

. l[d? \ f{x)ff{x){x-af \-[ 
when a is a pole of /(as). 



68] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 105 

Example 1. Shew that 



when — 1 < ^ < 1. 



^"sVr+^j'^iTid+s) "^ 27176 (i^-^^ "*■•••' 



Shew that the seoond member represents - , when | a? | > 1. 

Example 2. If S^^ denote the sum of all combinations of the numbers 

2«, 4«, 6«,...(2n-2)«, 
taken m together, shew that 

« sin«^^(2n-|-2)! l2n-|-3 2n-|-l ^-^ 3 p®"^^^ ' 

the expansion being valid for all values of z represented by points within the oval whose 
equation is | sin f | « 1 and which contains the point ;?=0. (Teixeira.) 

68. Expansion of a function of a root of an equation, in terms of a 
parameter occurring in the equation. 

Now consider the equation 

0{x)^(x-a)0i{x) = t, 

where ^ is a number such that along the contour 8 we have |^ (^)| > |^|> and 
along the contour s we have \0(z)\ < \t\. 

The equation (x) » t, regarded as an equation in x, will then have a 
single root in the ring-shaped region bounded by the curves S and s; we see, 
in £Btct, from the equations 

2m}a0W^t-2mlJsW) Js^) ^'"i 

-1, 

-0, 

that the equation in question has one root in the interior of S and none in 
the interior of s. 

Then if the function /(x) is regular in the region limited by 8 and «, we 
see from the preceding articles that the formula 

where An and Bn have the values already found, gives the expansion in 
powers of t of the function /(a?) of the root considered. 

As an example of this formula consider the equation {x - a) oosec x^^t, and let 

f(-r)=-i-. 
^ ' x-a 



106 THE PROCESSES OF ANALYSIS. [CHAP. VL 

Then we find 

. cosa 

" Sin a' 

*" (n+l)ln ^-+1 ' 

* sin a' * ' 



Hence 



cosa * t^ cP*+^(8in*a) . 1 



x-a sin a n-i(w + l)In cto»*i <sina' 



and thus gives the expansion, in ascending powers of t, of , where x is given in terms 

of t by the equation 

0? = a + < sin ^. (Teixeira. ) 

69. Lagrange's theorem. 

Suppose now that the function /(^) is regular at all points in the interior 
of £•, 80 that the poles 6n &s, ... &a do not exist. Then the formulae which 
give the quantities An and 5« now become 



1 d*-» 



m <"-'• 



^0 =/(a), 

Moreover the contour s can now be dispensed with, and the theorem of 
the last article takes the following form : 

Let /(z) be a regular function of z at all points in the interior of a 
contour fif, and let 6 (z) be a regular function with no zero in the interior of S. 
Let a be a point inside iS, and t a number such that for all points z on S wo 
have 

\(z-a)e(z)\>\t\. 

Then the equation (z — a)0 (z) = t will have one root x in the interior of 
S, and /(a;) will be given as a power-series in t by the expansion 



/(x)^f(a) + l -,J^ 



«— 1 



r. 



e^(ax 

This result was published by Lagrange in 1768 ; it is usually stated in a 
slightly different form, to obtain which we shall write 

the result may now be enunciated as follows : 

^//(^) ^^ ^ W ^ regular functions of z within a contour S surrounding 
a point a, and if the a quantity such thai the inequality 

t<l>{z)\<\z-a 



69] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 107 

18 satisfied at all points z on the perimeter of £>, then the equation 

z=sa + t<l>(z\ 

regarded as an equation in z, has one root in tlie interior of S: and if this root 
be denoted by a?, then any regular function of x can be expanded as a power- 
series in t by the formula 

This result is of course a particular case of the more general theorem 
given in § 68. 

Example 1. Within the contour surrounding ;e—a and defined hj the inequality 

|;?(«-a)|>|a|, 
the equation 

«-a— -=0 

has one root z, the expansion of which is given by Lagrange's theorem in the form 

Now jfrom the ordinary theory of quadratic equations, we know that the equation 

z—a — =0 
z 

has two roots, namely 

and our expansion represents the former of these ordy — an example of the need for care in 
the discussion of these series. If however we regard the expansion as a power-series in a, 
and derive other power-series from it by continuation in the a-plane, we shall ultimately 
arrive at the series 

« (-l)»(27i~l)! g'* 
»Ii n\ (n-l)I a*»-i' 

which represents the other branch of the function z. 

Example 2. If y be that one of the roots of the equation 

y^\-\-zf 

which reduces to unity when z is zero, shew that 



»(»+6)(w+6)(»+7) ^ w(w+6)(»+7)(»+8)(»+9) ^ ^ 

A. \ n \ 



41 ■ 6! 

so long as \z\<^. 

Example 3. If ^ be that one of the roots of the equation 



108 THE PROCESSES OF ANALYSIS. [CHAP. VI. 

which reduces to unity when y is zero, shew that 

the expansion being valid so long as 

\y\ < |(a-l)«-ia-«|. (McClintock.) 

70. ItoucM*s extension of Lagrange^ a iheorem. 

Consider now two functions /(^) and {z), which are regular at all points 



within a contour 0, on the perimeter of which the inequality j j;.\^ 



A") 

satisfied. 



< 1 is 



Then we shall shew that if the equation f(z)=:0 have p roots a^y a^, ... Op 
in the region contained by C, the equation f{z) — a<f> (z) = tuUl have p roots 
Oi', Oa', .•• ^'i *w ^f^ region; and for every function F(z) regular in the region 
we shall have 



r=l r=l n=l ^' r«l aO^ I -^^ (0.^)] ) 

f(z) 
where ylr (z) = ^ ' . 

We may note that this theorem reduces to that of Lagrange when 

f(z) ^z^a and p = 1. 

The result stated may be obtained in the following way : 
We have 2 J'(a,')-K^. f ^'/-^M^^^^dz 

{/(^)}'{/(^)-«<A(^)}J 



t/wl /(^) 1/(4 / (^) - «<^ w J ' 

When n is large, the last integral tends to zero: we thus have on the 
right-hand side a power-aeries in a, in which the coefficient of a* is 



or 



4 i r j^' \ F' (z) {<!> (z)]' jz - a,)"> [l 



r> ^ ■ ■ ■ ; I .'' J. t w m _^„ ^s: , ^j.i., . 1 j e^i^^mf^^m 



70—72] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 10& 



or I i JUl \ r (Or) 4> (ar)n 

which establishes the theorem. Putting F{z) = 1, it is seen that the number 
of roots a' is p. 

71. Teixeira has published the following generalisation of Lagrange's theorem, the 
proof of which may be left to the student. Let 

where ^ (z)^ ..,<t>k (z) are regular functions of z in the interior of a contour K, and ^ is a 
point inside K, Let a be a positive quantity, so small that the condition 



z-t 



z-t 



+ ...+ 



a*0fc {z) 



z-t 



<1 



is satisfied along the contour K, Then to every value of x which satisfies the condition 
I J? I < a there corresponds a unique value of « in the interior of K ; and / («), where / is a 
regular function at all points in the interior of JT, can be expanded in ascending powers of 
X by the formula 

where the summation is extended over all positive integral solutions of the equation 

a + 2^+3y+...+i{-X«n, 
and where 

Another form of this result is 

/W-/W +22 7-^ ^, {/' W *r-,. M (0}, 

M-K> v-0 syT*-) ' dt 
where the quantities <^v,^ are obtained from the equations 

72. Laplace's extension of Lagrange's theorem, 

Lagrange's result can easily be extended to a case in which the given 
equation is of a somewhat more general type. 

Suppose that the equation 

is given, and that it is desired to expand some function f{z) of a root of this 
equation in ascending powers of t 

If we write a + ^0 {z) = u, 

the equation reduces to i^ = a + <<^ {i/r (t^)}. 

The problem of expanding /(^) is therefore equivalent to that of expand- 
ing/{'^(w)}, where u is given by the last equation; and this can be done by 
Lagrange's theorem. 



110 THE PROCESSES OF ANALYSIS. [CUAP. YI. 

73, A further generoMsaiion of TayMs theorem. 

The series of Laurent, Darbouz, Burmann, etc. may be regarded as extensions in 
different directions of the fundamental series of Taylor. A generalisation of Taylor's 
theorem of a somewhat different character to these, is furnished by the foUowing result, 
the proof of which may be left to the student. 

Iff{z) and B {£) are regvlar functione oft in the neighbourhood of the point zt^x, and if 

6, ('^)^jj (t) dt, 3, W=//i W dt, 

and generally 

3n(^)=j\^iWdt, 

then, for values of tin the neighbourhood of the point x^ f{t) can be expanded in a series of 
the form 

/(«)«=ao^W+MiW+«2^2(«)+.-+«»^« («)+...> 
where 

and generally 

the number of differentiations in the last expression being n. 

It is clear that Taylor's series is obtained from this expansion by putting B {t)=h 
Example 1. Shew that 

Example 2. Shew that 

(Laurent, Joum. Math. Sp4c., 1897.) 

Example 3. By writing B {t)^e^, obtain the expansion of an arbitrary function of f in 
a series of the form 

where o^^, o^ are independent of t. 

Example 4. In the general result, shew that when ;r=0 we have 

where 



/W-2~5^ and d(«)-2^«*. 



^i^and d(^)-2^, 
ni ^ ' n\ 

(Quichard, Annates de Vic. Norm,, 1887.) 



78, 74] 



THE EXPANSION OP FUNCTIONS IN INFINITE SERIES. 



Ill 



74. The expansion of a function in rational functions. 

Consider now a function f(z), whose only singularities in the finite part 
of the plane are simple poles Oi, a,, Os, ...: let C], c^, ... be the residues at 
these poles, and let (7 be a circle of very large radius R not passing through 
any poles, so that f{z) is finite at all points in the circumference of C. (The 
function cosec^ may be cited as an example of the class of functions con- 
sidered.) Suppose further that at all points on the circumference of (7, the 
modulus of f(z) is less than M, where if is a quantity which remains finite 
when large values of R are taken. 

Then jr—, I J-^ dz^snm of residues of "^^ at points in the interior 
2w% JoZ — ic z -X ^ 

of (7 
=/(a?) + 2-^^, . 

where the summation extends over all poles in the interior of C. 

But ±( f<^dz = —(^-^^ + —.l -^^^dz 

iiriJce — m iirijo " 2in ] a z {z — w) 

-^ ^ ^ ^an 2inJoZ{Z'-x) 
if we suppose the function/(^) to be regular at the origin. 

Now R being supposed large, I -j^^—k is of the order -^ of small quantities, 

J gZyZ ^^X) JX 

and so tends to zero as R tends to infinity. 

Therefore on making R infinitely great, we have 

= /(.)-/(0) + 2c„(^-l), 



or 



m.m^%^U^*i\. 



which is an expansion off{x) in rational functions of ^. 

If instead of thd condition |/(«)| < J^ we have the condition \f{z) \ < MB^, where J^is 
finite for all values of R and n is a positive int^er, then we should have to ezpemd 

F{z)dz 



I 



o z—x 



by writing 






cosec;?: 



and should obtain a similar*but somewhat more complicated expansion. 
Example I. Prove that 

z ^ * \z-7iir nirj 
the summation extending to all positive and negative values of n. 

To obtain this result, let cosec z — =/(2)* The singularities of this function are at the 

z 

points issn^, where n is any positive or negative integer. 



■■■■JHW jjiwii j^.^^Mn^^PHPK^mnm 



^ 



112 THE PROCESSES OF ANALYSIS. [CHAP. VI. 

For points near one of these singularities, put ^smr + (1 Then 

/W=cosec(wir+f) l-^ilil^ ^ J- (l^-LY^ 



^ h positive powers of (. 



z — nir nir 



The residue of /(«) at the singularity nir is therefore ( - 1)*. Applying now the general 
theorem 

/(.)=/(0)+.c,[^ + I]. 
where c^^ is the residue at the singularity a^, we have 

/«=/(o)+s(-i)-j^-L_+i^}. 



But 



/(O) «» Lt,.o - + (positive powers of «) - - =0. 



Therefore 

cosecfs 



which is the required result. 

Example 2. If a is real and positive and less than unity, shew that 

6^ 1 « 2« cos 2nair - 4nir sin 2nan' 

For ^f{z)^- — r — , the singularities of f(z) are at the points zs^^nni, where 

n«=±l, ±2, ±3, ... ±00. 
For points z near z « 2wirt, put z = 2nirt + f . Then 

= — — +a series of positive powers of f. 

The residue at z^s^nin is therefore e*^'*. 

Also I • 

/(0) = fl±5±::.-n 

= [i(l+«.+ ...)(l+|+...)"-i2_^ 

Applying the general theorem 

/W=/(0)+2c,(^ + l). 

we have therefore 



H±«) ^i»a 00 sin2natr 
2 — T-^+ 2 



EXS.] THE EXPANSION OF FUNCmONS IN INFINITE SERIES. 113 

But 

- 2^. log ( - «-»^) = ^. (irt - 2a,r0 
1 

Thus 

«• — 1 z N.±i # — 27ii7r " «-i \« - 2mV z + 2niir) 
"* 2^ COB 2nafl' — 4n9r sin ^nair 

Example 3. Prove that 

1 ^_1_ _ 1_ I _2 1 

7r«*(fl«-2co8^+«~*)"2fra?* «*-«-* ir* + Jar* «*»-«-«» (2ir)*+i^ 

For the general term of the series on the right is 

{-\Yr _ 1__ 

which is the residue at either of the four singularities r, -ryvi^ -ri^ot the function 

vz 

(ir*a^ - JjF*) {eF* - e ~ »*) sin irz ' 

The singularities of this latter function which are not of the type r, -r, f% —riy 
are at the points 

^ ±s/i X ± sT-^i X 

V2 "• \/2 «■ 

2 
At «a=0 the residue is -^; 

at either of the four points «= ~ J" — , the residue is 

v2 'f' 

ir«~l 
/Vife _>/jr 



^ ) . VtJr 
S/ sin ._ 



V2 
Therefore 

__ 1 /" ffgcfe 

~ 2wt j c ('T*'?* - J^) (e^* - e - »«) sin irz * 

where C is an infinite contour. But at points on C, this int^prand is infinitely small 

compared with - ; the integral round C is therefore zero. 

W. A. 8 



114 THE PROCESSES OF ANALYSIS. [CHAP. VI. 

1 - (-lyr 1 :ii 






-1 



,^(;--)^_«'-"l}{.'-'>l _,»-«!} 



-1 



1 

ira^ (e« - 2 cos ^+«~«) ' 



which is the required result. 
Example 4. Prove that 

Example 5. Prove that 

coseoh*=l - Ste (^ - ^;^ + gj^ ...) . 

Example 6. Prove that 

8ech;r=4ir(^^,^^-^^^ + 2j^a^^...j. 

Example 7. Prove that 

coth *-l + to (^ + ^^-, + g^+ . ..) . 

Example 8. Prove that 



2 2 7—5- — o> , g . ,gv = -7 coth ira coth irb. 

(Cambridge Mathematical Tripos, Part I, 1899.) 

76. Expansion of a function in an infinite product 

The theorem of the last article can be applied to the expansion of 
functions as infinite products. 

For let / (z) be a function, which has simple zeros at the points Oi , a,, 0,, . . . 
where Limit | On | is infinite ; and suppose that f{z) has no singularities in 

the finite part of the plane. 

Then clearly /' (z) can have no singularities in the finite part of the 

f (z) 
plane, and 80*^-77^ can have singularities only at the places Oi, a,, a,, .... 



75] 



THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 



115 



Now for values o(z near Or, we have by Taylor's theorem 

/(z) = (^ - ar)f (a,) H- (l^f' (a,) + . . . 

and /' (^) =/' (a,) + (z^ Or)/" («r) + • • • • 

Thus we have 

:r7\ ~ ^ * constant + positive powers of (z — Or). 

J yZ) Z — Or 

f'(z) 
At each of the points Or, the function ''-^it^ has therefore a simple pole, with 

the residue + 1. 

f (z) . . 

If then -zj-r has at infinity the character of the functions considered in 

the last theorem, it can be expanded in the form 






Integrating this expression, and raising it to the exponential, we have 



where c is a constant independent of z. 



Putting 5 ~ 0, we see that/(0) = c, and thus the general result becomes 

f'io) 






This furnishes the expansion, in the form of an infinite product, of any 
function /(z) which fulfils the conditions stated. 

This theorem is a case of a general theorem on the factorisation of functions, which 
is due to Weierstrass, and which will be found in Forsjrth's Theory of Functions, 
Chapter v. 



Example 1. Consider the function /(«)< 



z 



, which has simple zeros at the points 



rn-, where r is any positive or negative integer. 

In this case we have /(O) =« 1 , /' (0) = 0, 

and so the theorem gives immediately 

sin 



^'-.?«{(-4)-l' 



since the condition relative to the behaviour of "^tt-t at infinity is easily seen to be 

fulfilled. 

8—2 



116 THE PROCESSES OF ANALTSia [CUAP. VI. 

Example 2. Prove that 

h©} {-(s.!.)} {-(s^.)} {-U^j} (-(^^1 

cosh k - COB X 
1 — oosx 

(Trinity College ExaminatioD, 1899.) 

76. Expansion of a periodic function in a series of cotangents. 

Another mode of expansion, which may be applied to periodic functions 
whose poles are all simple, is that indicated in the following example. 

Consider the function 

cot {x — Oi) cot (a: — a,) ... cot (x — On). 

This is a trigonometric function of a?, having poles at the points Oi, a,, ... On, 
and also at all other points whose afiSxes differ from one of these quantities 
by a multiple of tt. There is clearly no loss of generality in supposing that 
the real part of each of the quantities Oi, a,, ... 0^, lies between and tt. 

Now let ABCD be a rectangle in the j^-plane whose comers are the points 
J[(^ = — too), £(^sB7r— 100 ), C(j?=7r-»- too), and D(^ = too); and consider 
the integral 

^r — A cot (if — Oi) cot (if — Oj) ... cot (-^ — On) cot (j? — a?) ck 

taken round the perimeter of the rectangle. 

The integrals along DA and CB are equal but of opposite sign and cancel 
each other. Along CD, each of the cotangents has the value - 1, so the 

integral along CD is . Similarly the integral along AB has the value 

•5- . The whole integral has therefore the value 



l+(-l)\., 



%\ 



2 

The singularities of the integrand in the interior of the contour are at the 
points 2^ = Oi, Os, ... On, X ; and clearly the residue at tir is 

cot(ar — cti)cot(ar — aa)...cot(a,. — ar-i)cot(ar — Or+i) ••• 

cot (ttr — On) cot (Oy — x), 

while the residue at a; is 

cot(a: — Oi) ... cot(a? — ttn). 

Since the value of the integral is equal to the sum of all these residues, we 
thus have 

1 4.(- l)n • r=i» 

^— ^ t** = cot(a? — cti) ... cot (a? — On) + 2 cot(a,. — Oi) ... 

cot {Obr — a>f^ cot (Or — x). 



76, 77] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 117 

Thus if n be even, we have 
cot (a: — Oi) ... cot(a? — On) = 2 cot(ar — ch) • ••cot (or — On) cot (a? — Ctr) + (—l)*, 
and if n be odd we have 

cot(d? — Oi) ... cot(a?-a„)= 2 cot (ctr — cti) . . . cot (cv — On) cot (a? — a^). 

This method of decomposition into a series of cotangents is of very 
general application to periodic functions ; it may be regarded as the trigouo- 
metrical analogue of the decomposition of a rational function into partial 
fractions. 



Example. Prove that 

8m( 3? - ^i) sin (a? - fcj) • • • ^^^ (^ ~ K) ^^ (<*i "" ^i) • • • ^^^ (*i " ^1*) 
Bin (47 -a^) sin (a? -Oj) ...8in(a:-a„) sin (aj-Oj)... sin (aj-oj 



cot(4;— Oj) 



8m(aj-aj)...S]n(a2-aJ ^ 
+ 

+ C06 (a|+a2+...+a„-fci — 6j— ... — 6i»). 

77. Expansion in invei^se factorials. 

Another mode of development of functions, which although investigated 
by Schlomilch as long ago as 1863 has hitherto not been much used*, is that 
of expansion in inverse factorials. 

Let Z be a line drawn parallel to the imaginary axis in the j^-plane ; and 
draw a circle of large radius, having its centre at the point where I cuts the 
real axis. 

Consider a function f{z\ which has no singularities within the semi- 
circular area which is bounded by I and this circle and which lies on the 
positive side of Z ; let 7 be the semi-circular arc which bounds this region. 
Suppose moreover that at all points of 7 we have the inequality 

\f{z) i < M 

satisfied, where M is finite however large the radius of 7 may be chosen. 

Then if z be a point within this semi-circular region, we have 



Now 



[ f(t)dt ^r f(t)dt r zf(t) dt 



♦ Beferenoes to some recent work are given by Klajver, CampUs Rendui, oxxxir. (1902), p. 687. 



^^1 



— - ^> K* 



118 



THE PROCESSES OF ANALYSIS. 



[chap. VI. 



But 



/, 



zf{t)dt 

yt\t-Z) 



< \z 



m\ 



dt 



y\t\\t-z\' 
which is infinitesimal when the radius of 7 is infinitely great. 

Thus 






if we now suppose that the direction of integration along I is from — loo to 

+ 100. 

Now if n be any positive integer and z be not equal to 0, — 1, — 2, etc., 
we have the identity 



1 1 



+ ...+ 



z-t -^ • -8^(^+1) ' ^(^-i-l)(^ + 2) z{z-k-l)...{z + n){Z'-ty 

on substituting this in the second integral we have therefore 



Oi 



y(-^) = ao + ^ + 



(h 



z ziz-^-l) 



+ ...+ 



^'Wi 



where 



z{z-k-\) ,,.{z -^-n) 
■^ 2in}iz (z + 1) ... IzJf n) (z - 1) ' 



/(t)dt 
t 



^=2-^//(^>^^' 
Now the product 



can be written 



t(t+l)...(t + n ) 
z{z-{'l)...(z + n) 



t ** 

-n 



r 



and it diverges to zero or to infinity when z tends to 00 according as the real 
part of t—z is negative or positive, as can be seen by comparing it with 
the product 



77] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 119 

which has the value (n+ iy~'. But the real part of ^ — <« is, in the case 
under consideration, negative ; and so the product 

t(t+l)...(t + n) 

is infinitesimal when n is infinite. 

Since/(^) is finite along I, and / . *^' . is finite, we see that 



I z{Z'\-l)...(Z'^ n){z^t) 
is infinitesimal when n is infinite. 



We can therefore expand f(z) in the form 

>^<^>="«+ 7 + ^(^Ti) + ^(7+lHl+ 2) + - • 

the coefficients a being given by the above equation ; and this expansion is 
valid for all values of z whose real part is greater than the real part of z at 
any of the singular points of f(z), except for the points 

^ = 0, -1. -2 

Example 1. Obtain the same result by using the equalities 

«(«+l)(f+2)...(«+n) n\Jo ^ «/-««, , 

Example 2. Obtain the expansion 

where a„= r<(l-<)(2-<) — (^-l-0«fe> 

and discuss the region of its convergency. (Schl6milch.) 

Miscellaneous Examples. 

1. Let er^Pn denote the nth derivate of e'^y so that 

P^,= l, P,= -2«, P,-4a»-2, etc 

Shew that if f(z) is an arbitrary function, then/(«) can be expanded in the form 

1 r* 

where a-= | e-^Pn(x)f(x)dx. 

2.4.6... 2nVir; — 

and find the region of convergence of this series. (Hermite.) 



120 THE PROCESSES OF ANALYSIS. [CHAP. VI. 

2. Obtain (from Darbouz's formula or otherwise) the expansion 



+ ; 

find the remainder after n terms, and discuss the convergence of the series. 

3. Shew that 

+( - D— ^•^•°-;-,j^""^^ J {/" (*+ A)+( - !)«/• (*)} 



( - l)»h<'*' j\,{t)f*'{x+/U) dt, 



where y, (*)= -^^i-^ x»+i (\-x)n+i^{x-i (1 -a:)-*} 



rrfilj 



and shew that y^ (x) is the coefficient of n ! ^ in the expansion of {(1 — tx) (1 +<— tir)}'i in 
ascending powers of t. 

4. By taking 

in Darboux's formula, shew that 

/(*+A) -/(x)= -a,h {/' (*+A)-l/' (*)| 






V 1 — r w tt' u' 



MISC. EXS.] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 121 

5. Shew that 



2! 



~ ^^»^^;P<'-° ^r'(a)4-/-(.)} 



+ 






271 ! 



JV2m(0/<*'**U«+<(^-«)}<'^ 



-^ ♦.<')%-|r[^^'.(.t;)]...- 

6. Prove that 

+ (7«(«,-^)V'M^)+ 

where CJ^ is the coefficient of 2!* in the expansion of cot (^-5)1^ ascending powers of z, 

(Trinity College Examination.) 

7. If ^1 and x^ are integers, and <^ (2) is a function which is regular for all values of z 
(finite or infinite) of which the real part lies between x^ and x^, shew (by integrating 



/ 



4>{z)dz 



£>2iriz _ 1 

round a rectangle whose sides are parallel to the real and imaginary axes) that 
i*(*'i)+*(^i + l)+<^(a?i + 2)+...+<^(a?,-l)+i<^(^^ 

= ['*<f>{z)dz'k-l /'" ^(^tH-<y)-»(:Ci4-*y)-»(^,-ty)+»(j?,-ty) ^^ 

Henoe by applying the theorem 

where i?i, ^3, ... are the Bemoullian nimibers, shew that 

^(l)+<^(2)+...+<^(n)-C+i^(n)+|**(«)c?^+^|^^-^^^^^ 
(where (7 is a constant not involving n) provided that the last series converges. 
8. Obtain the expansion 

for one root of the equation x^2u+u\ and shew that it convei*ges so long as | :p | < 1. 



122 THE PROCESSBS OF ANALYSIS. [CHAP. VI. 

9. If /S^i, denote the sum of all combinations of the numbers 

1«, 3«, 6«, ... (2n-l)« 
taken m together, shew that 

* ~8in»^^(2n+2)! l2«+3 "ifn+i) 2»+l* •" ^ »(»+« Sj 

10. If the function /(<) is r^[ular in the interior of that one of the ovals whoee 
equation is | sin2|>'C (where C^ 1), which includes the origin, shew that /(<) can, for all 
points < within this oval, be expanded in the form 

- /P»)(0)+;S*2/(*-»(0)+...+^"-»/<'>(0) 

n')-m^^r — 2^1 '^•"' 

. g /'--''(0)+^^,./O'-')(0)+...+C/'(0) ^^...., 

where ^S^' is the sum of all combinations of the numbers 

2«, 4«, 6«, ... (2n-2)* 

taken m together, and £f^^^^ denotes the sum of all combinations of the numbers 

1«, 3«, 6«, ... (2n-l)«, 
taken m together. 

11. Shew that the two series 

2f^ 2^^ 



d 2g 2 / 2g Y 2 . 4 / 2^ y 

represent the same function in one part of the plane, and can be transformed into each 
other by Burmann's theorem. 

12. If a function f(z) is periodic, of period 2fr, and is regular in the infinite strip of 
the plane, included between the two branches of the curve | sin «|kC7 (where 6'>1), shew 
that at all points in the strip it can be expanded in an infinite series of the form 

/($)= Aq-^- Aisin z-^ .,, + A^ain^ z-^ 

+coe «(i9i+5j sin «+...+5i»8in«-i «+...); 

and find the coefficients A and B. 

13. If <f> and / be connected by the equation 

of which one root is a, shew that 



^W ^ 1 ^^'•'^ ^2! <^'» <1>"{PF')' 



1 
1 



^ JL 
31 <^'« 



4>"' (*T' ( /'^T 



1.1.2 



T .••*.•, 



MISC. EXS.] THE EXPANSION OF FUNC?riON8 IN INFINITE SERIES. 123 



where Fy /, F\ etc. denote 



F(a\ f{a\ -^- 



14. If a function If (a, 6, x) be defined by the series 



IF (a, 6, ^)=a:+ 0-7-^+ 



which converges so long as 



2! 



x\< 



31 



*•'+ 



h\: 



shew that dx^^""^ *' ^)-l+(a-fc)ir(a-6, 6, d?); 

and shew that if y « ir(a, 6, x)y 



then 



a;» IT (6, a, y). 



Examples of this function are 

ir(l, 0, ^)=^-l, 

If (0, 1, a:)=log(14-a;), 
(l + ;c)«-l 



15. Prove that 



If (a, 1, 0?)^ 



1 1 . ; (-ir^ g 



00 

n—O 



where 



Q. 



4a^ 



3a, 
6a, 





4a| 







3ao 









(2n-2)aH-i : (n-l)ao 



na^ 



(n-l)a^., a. 



and obtain a similar expression for 



< 2 a,idf*> . 
(n-o ; 



(Jeiek.) 



(Mangeot.) 



16. Shew that 



2 CLfSf 







where /S^ is the sum of the rth powers of the roots of the equation 



2a,.a?*'«*0. 




(QambrolL) 



17. If /»(«) denote the »th derivate of /(«), and if /-»(«) denote that one of the nth 
integrals of f{z) which has an n-ple zero at < *0, shew that 



and obtain Taylor's series from this result, by putting g (s}b1. 



(Quichard.) 



124 THE PBOCESSES OF ANALYSIS. [CHAP. VI. 

18. Shew that, if ^ be not an integer, the series 



2 



(xH-m)*(arH-n)*' 



in which m and n receive in every poesible way unequal values, zero or integers lying 
between +/ and -/, vanishes when /increases indefinitely. 

(Cambridge Mathematical Tripos, Part I, 1895.) 



19. Sum the infinite series 

1 

n— 






where the value n=0 is omitted, and p, q are positive integers to be increased without 
limit 

(Cambridge Mathematical Tripos, Part I, 1896.) 



20. If F(x)^ei^'"^^'^^^, shew that 






n-l 

and that the function thus defined satisfies the relations 

F{x)F(l-x)^2amxw. 
Further, if ^(r)=:«+^ + ^,+ 



-J^og(l-«)rf(log4 



shew that /'(ar)=c*'^2^*<^~'''*^^ 

when I !-«-«»*« I <1. 



(Trinity College Examination.) 



21. Shew that 



[-(r][-(i^)l-G^J][-G-^J][-GA^)"] 



n 

I 

2«(1-C08A')*«" * 



where (Lf^Jc Bin -^ — ir, 

n 



MISC. EXS.] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 125 

" n 

and < j: < 2ir. (Mildner.) 

22. If I ^|< 1 and a is not a positive int^er, shew that 

where (7 is a contour in the ^plane enclosing the points 0, x, (Lerch.) 

2a If ^(«), <^,(«), ... are any polynomials in z, and if F(s) be any function, and if 
^1 (')) ^a (^)) ••• ^ polynomials defined by the equations 

[* ^(4r) ,^, (*) *, (X) ... <^,_, (*) *"> )-»» W d«=^„(,), 

y a a— ar 

shewthat P ^W^ ,^iW, ^^aW 



*i W 4*% («) *s W 
1 



rF(x)<h{x)<h(x).,.4,^{x)^. 

J a t—X 



24. A system of functions p^ (z\ p^ (z\ p^ (z), ... is defined by the equations 

where a» and b^^ are given functions of n, which for n^oo tend respectively to the limits 
Oand -1. 

Shew that the region of convergence of a series 

where e^y e^y ... are independent of «, is a Cassini's oval with the foci +!> - 1. 

Shew that every analytic function f(z\ which is regular in the interior of the oval, can 
for points in this region be expanded in a series 

where 






126 THE PROCESSES OF ANALYSIS. [CHAP. VI. 

the integrals being taken round the boundary of the region, and the functions q^ {z) being 
defined by 

26. If Pn (*) be the coefficient of — j in the expansion of 






in ascending powers of z, so that 



shew that 

(1) Pn {x) is a homogeneous polynomial of degree n inx and A, 

(2) 2-^-- (^^^)' 



(3) 



rPn{s)dx^O (n^l), 



(4) If y — o^Po (x)-^aiPi (^) H-OjPj (*) + .. ., where a^, o^, a,, ... are real constants, 
then the mean value of -j^ in the interval from x^ - A to x^m 4.A is o^. (L^ut^) 

26. If P« (j?) be defined as in the preceding example, shew that 
/>^-(-l)-2^^(^cos-j-2,^cos-^+3^cos-^ +...J, 

i'2»+i = (-ir2^^:^j(^sm-^-2^^jSm-^ + ^5;^jSm-^ + ...j. (AppelL) 



i 



CHAPTER Vll. 
Fourier Series. 

78. Definition of Fourier series; nature of the region within which a 
Fourier series converges. 

Series of the type 
Oo + a, cos ^H-ag cos 2-^ + 08 cos 3£r + ... + 6i8iiiz + 6,sin 2^ + 63sm 3£r + ... , 

where Oo, Oi, a,, a,, fej, ftj, 6,, ... are independeat of z, are of great import- 
ance in many analytical investigations. They are called Fourier Series, 

We have already seen that the region within which a series of ascending 
powers of z converges is always a circle \ and the region within which a 
series of ascending and descending powers of z converges is the ring-shaped 
space between two circles \ we are therefore led by analogy to expect that 
series of the Fourier type will likewise converge within a region of some 
definite character. 

To investigate this question, write 6**= f. 
The series becomes 

This is a Laurent series in f ; it will therefore be convergent, if at all, 
within a ring-shaped space bounded by two circles in the ^-plane ; that is, 
it will be convergent for values of f satisfying an inequality of the type 

• 

a<|?|<fe, 
where a and h are positive constants. 

Now let 

£r = a: -h ty ; 
then 



128 THE PROCESSES OF ANALYSIS. [CHAP. VU. 

aod therefore the inequality becomes 

log a< — y< log 6. 

This inequality defines a belt of the ^r-plane, bounded by the two lines 
y = — log a and y = — log h ; hence the region of convergence of a Foui-ier 
series is a belt of the z-plane, bounded by two lines parallel to the real axis. 

It may however happen that the Laurent series in f is divergent for all 
values of f, in which case the Fourier series is divergent for all values of z ; 
or, (and this is the most important case for our purpose,) it may happen that 
a » 6, so that the region of convergence of the Laurent series narrows down 
to the circumference of a single circle in the ^-plane ; in this case the region 
of convergence of the Fourier series narrows down to a single line parallel to 
the real axis in the plane of the variable z. 

If now the coefficients Oq, Oi, a,, ... 6], &t, ... are all real, considerations of 
symmetry shew that if the Fourier series is divergent for a value z^a + tb, 
it will also be divergent for the value -zr =s a — i6 ; so if in this case the region 
of convergence narrows down to a line, that line can only be the real axis in 
the ^-plane. 

Hence a Fourier series with real coefficients may converge only for real 
values of z, and diverge for all complex valves of z. 

An example of this class of expansions is afforded by the series 

sin-^ — 2®^^2^+3^^3^"" 4 s^^ 4^+ ... . 
Writing this in the form 

we see that it diverges when z is not purely real ; when z is purely real and 
not an odd multiple of ir, the sum of the series is 

ilog(l+6'0-^.log(H-6-^X 

or 21'^^^**' 

or i^z-Vkir, 

where k is some integer, as yet undetermined. 

Now when z = the sum of the series is seen directly to be 0; when 

z^-ci. the sum of the series is tan~^ 1, or 7- ; when ^ = — — the sum is 
2 4 2 



78] FOURIBR SERIES. 129 

— tan""* 1, or — 7 . In this way we see that when z lies between — ir and + ir, 

the integer k is zero. 

But A? is no longer zero when z is greater than ir ; for each term of the 
series is clearly unaffected if ^ + 27r be written for z : hence the sum of the 
series must be the same for -^ + 27r as for ir ; and hence when wkzk Stt, 

the sum of the series is ^z-^tt; so that when z lies between ir and Stt, the 
integer Ar w — 1. 

Proceeding in this way, we see that the sum of the Fourier series is 
^z + kir, where k is an integer chosen so as to make ^z + kir lie between 

— ^ and + ^ . This is important as shewing that the sum of a Fourier series 

is not necessarily a continuous analytic function. It is clear however that the 
sum of a Fourier series can have discontinuities only in the case in which the 
region of convergence narrows down to the real axis ; in the other case when 
the region of convergence is a belt of finite and infinite breadth, the Laurent 
series in ^ represents an analytic function, and therefore the Fourier series in 
z does also. 

Example. Shew that the series 

cos «- 52 cos 2« + 5j COS3-5- ... 

n^ 1 
converges only for real values of 2, and that when -ir<«<-firit8 sum i^ ts - 7 **• 

For when z is real, the series is absolutely and uDiformly convergent, as is seen by com- 
paring it with the series 1 + 02 + 02 "*"••• • 

When z \a complex, we have (putting z=x+iy) 

-ijC0srw=^ {e*(»«+"*»')-|-e<(-»«-»«*')} ; 

now either — , or ^ s ^ infinite for n=ac, so the terms of the series are ultimately 

infinitely great and the series diverges. 

To find the sum when z is real, it has been shewn that when -7r<z<ir we have 

^z=:8in z- ^ sin 2i+i sin 3;? ... . 

This series is imiformly convergent in the interval (though not at its extremes - ir and fr) 
and so can be integrated. 

Thus c-J«*=cos;8-5jCos2a+^cos3«-,.., 

where c is a constant. 

W. A. 9 



130 THE PROCESSES OF ANALYSIS. [CHAP. VII. 

To find c put «»0, which gives 



whence the result. 



,1.1 _^ 
^" 2«"*"3«"*""12' 



79. Valties of the coefficients in terms of the sum of a Fourier series, when 
the series converges at all points in a belt of finite breadth in the z-plane. 

The connexion between the coefficients a^, Oi, a,, ... , 6i, 69, ... of a Fourier 
series, and the sum of the series, can be easily found in the case in which 
the series converges in a belt of finite breadth in the ^-plane. For in this 
case, as we have seen, the sum of the series is an analytic function of z. Let 
it be denoted hyf{e), so that 

f(t) ss a^ + a^cos z + a^coa 2z + ... +6, sin 5-^69 sin a? + — 
Writing f = e^, the series becomes 

and by Laurent's theorem the coefficients in this expansion are given by the 
equations 



2^^-^^j /(^)?-»d?. 



where C is any circle in the (-plane, surrounding the origin and contained 
within the ring-shaped region in which the expanded function is regular. 
Now if the quantities Or and br are all real, we see as ^before by symmetry 
that the real axis must be contained in the region of convergence in the jr-plane, 
and therefore the circle of radius imity must be contained in the region of 
convergence in the (-plane, since this circle corresponds to the real axis in the 
jff-plane. We can therefore take (7 to be a circle of radius unity, with the 
point (b as centre. 

Now writing f « e^ in the integrals, we have 



and so 



(ar + ibr)^j^y(z)(f^dz, 

1 r*» 

ttr = - / f(z) COS rzdz (r > 0), 

1 ri» . 

6r « — I f{z) sin rzdZf 



79, 80] FOURIER SERIES. 131 

and ^"""^ZttJ f^^^^' 

These equations give the vahies of the coeflScients Oo, Oj, a,, ... , 
bi, ftj, ... , of the Fourier series, in terms of the sum f(z) of the series, 
in the case in which the series converges over a belt of finite breadth in the 
2:-plane. We shall see in the next article that the same formulae hold good 
in the more extended case, in which the series converges only for real values 
of z. 



sme 



JBaxunpU, Shew that the function , f^ can, when k<l, be expanded in a 

Fourier seriee of sinee of multiples of «, valid for all points z situated in a belt, of width 
-2 logir, parallel to the real axis in the ;B-plane. 



For we have 

sin a 



= Li_i 1 .1 



00 I 

and this can be expanded in the form 2 i, /;* (6***-e-**»), provided k le**! and ;t |e-*»| are 
less than unity. This can only happen when their product X^ is less than 1, Le. when 

-l<it<l. 

When this condition is satisfied, on putting z^x-^iy, it is clear that we must have 

e*«-i'|<T and>it, Le. we must have-y lying between logfi) and log it, Le. z must be 

within a belt of width -21ogiI;, parallel to the real. axis. When these conditions are 
satisfied the expansion is valid, and so 

8in« * . , . 

2 A!* *smn«. 



l-2irco8;j+it* 



Hal 



80. Fourier's Theorem. 

ft 

We have already said that the most interesting cases of Fourier's series 
are those to which the investigation of the last article cannot be applied, 
on account of the fact that the series converges only for real values of z. It 
is therefore necessary to undertake another investigation, in which the 
assumptions of the last article are no longer made. The result to which we 
shall be led is known as Fourier's theorem, and may be stated thus : 

Iff (z) be a quantity which depends on a variahle z, and which is finite and 
has only a limited number o/maaima and n^inima and of finite discontinuities 
in the interval <z< 2ir, then the sum of the series 



CO 

^0+ S (amOosmz + bmsiamz), 

mai 



9—2 



132 THE PROOBSSES OF ANALTSia [CHAP. VII. 

where 

1 f *» 
dm = - / /(t) COS mtdt, 

1 r*» 

6m= - / /(O sin TYVtdt, 

repre^ento y (2:), at every point in the interval < ^ < 2^ for which f(z) is 
continuous; and at every point in the interval 0<z<27r for which f{z) is 
discontinuov^Sy the sum of the series is the arithmetic mean of the two values of 
f(z) at the discontinuity. 

The diacussioQ of Fourier's theorem given below is a modification of what is known as 
Cauch^i second proofs which was originally published in 1827 in the second volume of his 
Exercices de Math^moOiqueif and is reprinted in his Collected Works, Second Series, YoL vn., 
p. 393). 

This proof, (which in its original form was in some respects imperfect,) seems to have 
been little used by the mathematicians of the nineteenth centiuy, who in the discussion 
of FourieHs theorem almost universally followed the exposition of Dirichlet (which is 
also reproduced later in this chapter) ; the importance of Cauchy's proof was shewn by 
A. Hamack in 1888. It may be observed that the restrictions placed on /(«) — as to its 
having only a limited number of maxima and minima, etc. — are suficieni but not necessary 
for the validity of the expansion. 

To establish the theorem, we write the first 2ik + 1 terms of the expansion 
in the form 

~l f(t)dt'^^ 2 f{t)coam(Z''t)dt, 



or 



or . Uk-h Fjk, 

where Um = "*2* ^ f V* <*- V(0 dt, 

We shall now investigate the behaviour of the quantity J7fc when k, though 
finite, is a large number. 

Let <t> (^) denote the quantity 



80] 



FOURIER SERIES. 



133 



Then <f> (^ clearly has a definite value corresponding to every value of ^, 
except the exceptional values ^=0, ±i, ± 2t, for which c*^= 1 ; moreover, 
it is easily seen that the quantity 

tends to a definite limit when B^ tends to zero, independently of the way in 
which Sf tends to zero (still excepting the points 0, ±i, ±2i...); hence 
<f>{^) is an analytic fiiDctioD of f, having poles at the points 0, ± t, ± 2i ... ; 
and the series U^ is clearly the sum of the residues of ^ (^) at those of its 
poles which are contained within a circle C> in the ^-plane, whose centre is 

at the origin and whose radius is (* + 2) • 



Hence 



Write 



Thus 






Now we can write 



.i-»-» 



+»-» j^.i-i »jr+»-i 



ir , , _ 



^*==2Uf +f +f +r +r u*(^^^ 

«/i + /, + /3 + /4 + /b say. 

At points in the range of integration of /, and I^, the real part of ^is 
positive and at least of order k'^; and so for these integrals we have 

In this expression, as k tends to infinity, ^^-^^^ tends to the limit unity, 

and {'e^(»-<-*r) tends to the limit zero : thus ^<f>(^ tends to the limit zero, 
since the range of integration from to ^ is finite ; and hence as /i and I^ 
are the integrals of ^(f> ((^ taken over finite ranges, we see that /i and I^ tend 
to zero as k tends to infinity. 

Considering next the integrals /, and /«, we observe that the quantity 
^^_^ is never infinite when < ^ < 2: < 2'jr, and so f^ ((^ is never infinite ; 



134 THE PROCESSES OF ANALYSIS. [CHAP. VIL 

and thus, since /{ and /« are integrals of ^(f> (^ over ranges which become 
infinitesimal as k tends to infinity, it follows that /, and /« tend to zero as k 
tends to infinity. 

Consider next the integral /j, or 

where f* (?) « ^^jjeii^t,f(t) dt 

In the range of integration of /«, the real part of ? is negative and at 

least of order W. The feu^tor -r-? — r- therefore tends to the value — 1 as fc 

6*^— 1 

tends to infinity ; denote it by — (1 + aj^), where o» tends to zero as k tends 

to infinity. 



Now r^(^^*^/(t) dt^J, + /„ 

^0 



where /,«[ ^'^ l:^^'^fit)dt, 

and t7,= r ^ K^^-^f{t)dL 



'"log* 



Considering first /i, we see that within its range of integration the 
quantity ^{z-^t) has its real part always negative and at least of order 

'. — T, which tends to infinity with k\ hence the quantity ^ef^^^^ tends to 

zero as k tends to infinity ; and therefore as the range of integration in t/^ is 
finite, we see that Ji tends to zero as k tends to infinity. 

Consider next J^. Writing t; ■» ? (-«r — e), we have 

log* 


and writing e^^Wy this becomes 



/•log* / ^\ 



J.^j^''f[z-'2^)d.. 



Now as k tends to infinity, 6*^* and — ^ tend to the limit zero. Let 
/(^— 0) denote /(^) if ir is a point at which the function f{z) is a continuous 



80] FOURIER SERIES. 135 

function, and at those points at which f(z) is a discontinuous function let 
f(s — 0) denote that one of the two values oif{z) which is continuous with 
the value of/ for values smaller than z. Then since there cannot be another 
discontinuity within an infinitesimal distance of z, we can write 



/(-^)'/(*-0) + ,. 



r 

where 17 tends to zero aa k tends to infinity ; and so 

f -L. 



[e fe 

t7,=/(^ — 0)1 dw+l lydiw; 



jf - log* 

= -/(^ - 0) + e'^^'fiz - 0) + j ^ vdw, 

or J,=:-/(^-0) + €, 

where e tends to zero as k tends to infinity. 

Thus 

l;<l>(0 (l + «*){/i-/(^-0) + £}, 

where a», Ji, and e, each tend to zero as k tends to infinity. We can write this 
^<f> ((^ ^/(z — 0) + T, where t tends to zero as k tends to infinity ; and this is 
true throughout the range of integration of the integral is- 

Thus 
or /, = 5 /(z — 0) + a, where <r tends to zero as k tends to infinity. 



Hence finally 

I7» = 5/(-^-0) + <r+/, + /. + /4 + A, 

where <r, /i, /,, /«, I^ each tend to zero as k tends to infinity ; which can be 
written 

where Uj^ tends to zero as k tends to infinity. 
Similarly we can shew that 

where v^ tends to zero as k tends to infinity, and where /{$ + 0) denotes f(z) 



136 THE PROCESSES OF ANALYSIS. [CHAP. VH. 

if 2^ is a value for which the function/(^) is continuous, and denotes the value 
oif{z) for values slightly greater than 2: if 5 is a value for which the function 
f{z) is discontinuous. Hence the sum of the first (2A; + 1) terms of the 
Fourier series is 

\f{^ - 0) 4- ^/(^ + 0) + w* + V*, 

where uj^ and v* tend to zero as k tends to infinity ; the sum to infinity of the 
series is therefore 

^{/(^-0)+/(^ + 0)]. 

which establishes Fourier's theorem. 

It must be observed that the sum of the series coincides with f{z) only 
far values of z between and 27r ; outside these limits the sum S (z) of the 
series can be found from the circumstance that S{z-\'2nir) = S(zX (a result 
which is obvious, since all the terms are periodic) ; while f{z) may of course 
have any values whatever when z is not included between the limits and 2^, 

Example, Take a function /(a) such that 



IT 



f{z)wm- from «««0 to «=ir, 

and /(«) = -J from;? « IT to «=2ir. 

The corresponding Fourier series is 

Oo + SOm OOS97U + 26m siu mz, 



where O'm^^- \ f(t)co3mtdt, 



n J 



'm 



1 r«» 

-i /(Osi 
irjo 



sin mtdt, 



These integrals give 

1 f» 1 f^ 

00=0, Otn'^i I cosmtcfe-- / cosmtdt^'Oj 

6«=T / sinm^(ft-7 I nin mtdt=-r- (I -cos mw). 
4;o 4j» 4m'' 

Therefore 6»,=0 if wi ia even, and 6„»= — if m is odd ; and so we liave 

J, . sinz sindi? . sinSz . 

which is the required Fourier expansion. 



81] FOURIER SERIES. 137 

This series can be summed by elementary methods in the following manner. We 
have 

. ^sinat^sin52 1 ( ^^^ ^ \ 1 / ^^«-w^ \ 

4i*^(l-e<')(l+fl-^) 41^"^ 4+ 2 » 

where r is an undetermined integer. It is clear from the above that r actually has the 
value zero when < 2 < ir, and unity when w<z<2w. 

81. The representation of a function by Fourier series for ranges other 
than to 2^. 

Suppose now that the range of values of z, for which it is required to 
represent a function f(z) by a Fourier series, is not the range from to 2^, 
but from a to b, where a and b are any given real numbers. To extend 
Fourier's result to this case, we take a new variable z defined by the 
equation 

and write 

6 — a / 



•^(«+-^^)'^<'^- 



Then F{/) is a function whose value is given for all values of its argument 
y between and 2^. 

Therefore by the previous result we have 

^(i^ = A fV(Od^+- 2 f*'cosm(/-0^(0^', 
or writing 



we have 



6 — a , 



/«-^J>)*-E^,.!j> ^™t5-/«^ 



This last result may be regarded as the general form of Fourier's theorem. 
• Example. To express the function — ^ — ^^ as a Fourier series, valid when 

-ir<z<ir. 

Here a™ — ir, 6«»r. 



138 THE PROCESSES OF ANALYSIS. [CHAP. VII. 

The formula therefore becomes 

/W-i r /(0^+; 2 f' coBn{z-t)f(t)dt. 

^*r y -r w nail J -» 

Since in this case / (O— -/( - 0» ^^ reduces to 

/(«)=- Z sin Yu / — --sinn^(i< 



•■ 2 sm nj I ^ . . ^i _^ ~ dt 



ii.iirt («**»- e~"^) I m+tn m-in J 

• (-l)»sin nz / 1 _ 1 \ 



HBi »r(m'+n«) ^ 
which is the required expansion. 

82. The Sine and Cosine Series, 

We proceed to derive two particular cases of Fourier's theorem which are 
of frequent occurrence. 

Suppose that a function f(z) is given for a range to 2 of values of the 
variable z, and that we require a series which shall represent f(z) for these 
values of z^ and which shall have the value /(— z) for values of z between 
and — L 

To obtain a series of this character, we write in the preceding result 
a » - Z, h^ly /{—z) ^f(z). Thus we have 

/(*) - ^ijjit) dt + ] J J'^cos^'^-^l^- *>/ (*) dt, 



/ w - i/Vw '^ + T A «» ^£ «» ^y (0 dt. 

which is called the Cosine Series. 

If on the other hand we require a series which shall represent /(^) for values 
of z between and I, and shall have the sum — /(— z) for values of z between 
and — I, we write in the general result a = — Z, 6 = hfi" z) =» — /W> and 
thus obtain 

f(z) « 7 2 sm -J- I sm j-f(t) at, 

^ m-l f' Jo ^ 

which is called the Sine Series, 



82] FOURIER SERIES. 139 

If ^ z 

EsxanpU 1. Expand -—^ sin « in a cosine series, valid when 0<f <fr. 
When < f < IT, we have by the formula just obtained 

-sinf-s^- I (w-t)smtdt+- 2 cosmi / {w''t)BmtooenUcU 
«r y IT iM>i J 



2 



•i[o"^'"~']"i/o'^^^ 



1 • 

+ 5— 2 OOSf/tf 



|r (fr-08in(m+l)*(ft- j (ir-t)mn{m-l)tcu\ 

1.1 ^1 1 r ^ ^ 1 

■■S + l<5<»'+S- 2 cos HI* >-= y 



11 * cosms 

5 + 70081- 2 



2^4"^' ^,(m-l)(m+l)- 
The required series is therefore 

It will be observed that it is only for values of « between and w that the sum of this 
series is proved to be —^ sins; thus for instance when « has a value between and — ir, 

the sum of the series is not ^^^— sin^ but - ^5— sin g; when « has a value between ir 

2 2 

and 2ir, the sum of the series happens to be again — ^— sin s, but this must be regarded 

as a mere coincidence arising from the special function considered, and not from the 
general theorem. 

Example 2. To expand ^-g— - ^ * «^« series, valid when 0<«<ir. 
We have 



— ^ — - ■■ — 2 sm 9IU I — ^-5 — •' am nUat 
o n fli*i Jo o 



■■ 2 smini J -^— ; — ^ BinnUdt. 

Mai 



'io-4 

r«(ir-20ffln««n ^ 1 /■» . ,j. 



Therefore 



2m« 
vziw'-t) . sinSf . sin5f . 



140 THE PB0CES8ES OF ANALYSIS. [CHAP. VTI. 

Here again the sum of the series is -^ — ^ only when z lies between and n. Thus 
when s lies between «• and 2»r, the sum of the series is — ~ — . The sum of the 

o 

series for values of z beyond the limits and ir can be found at once from the equations 
S (t)=» - S {" z) and S{z+2ir)^S(z)y where S(z) denotes the sum of the series. 

Example 3. Prove that, when < « < ir, 

yr (tr ~ 2z) ( w' + 2nz - 2^) cos3« cosSa 

96 -cosa+-g^ +__+.... 

For when < « < «• we have 

— ^ ^-i-jr^ i«- 2 cosm« I -> ^-^ "^cosm^cb 

96 fTfwsi Jo 96 



(integrating by parts) = 2 cosm« I sinm^cfe 

■1=1 J 4^ 

(integrating by parts) = 2 cos me J ~-n- cos mtcft 

m=i Jo 4m' 

» /■» 1 

(integrating by parts) = 2 cosm;; ) r— .sinmtcfo 

mat y *^ 



« i-(-i)- 

* — s — 2 — oosm* 
m-i 2m* 



. cos Zz . cos 5e 
eoe f +— ^ + -^j~ + ... . 



Excm/ple 4. Shew that for values of z between and ir, «** can be expanded in the 
cosine series 

and draw graphs of the function «** and of the simi of the series. 

Example 5. Shew that for values of z between and ir, the function — ^^-^ — - can 
be expanded in the cosine series 

. cos Sz . cos 52 

and draw graphs of the function -^-^ — < and of the sum of the series. 

83. A Itemative proof of Fourier's theorem. 

Another proof of Fourier's theorem, based on an entirely different set of ideas, is due 
to Dirichlet*. 

♦ CoUecUd W&rke, Vol. i. pp. 188—160. 



83] FOURIER SERIEa 141 

CoDsider first the sum of a limited number of terms of the series 



00 



where 



a^^- lf{t)coafntcU (m«l, 2, 3, ...), 

IT J 

6^=i [^/(Osinm^cft (m-1, 2, 3, ...), 

V J 

and where z is supposed to be a real variable. 
Since 

V J 

we have the sum to (2m +1) terms of the series expressed by the formula 
^m-- /{i+coe(^-«)+cos2 (<-«)+. ..+oosm(<-«)}/(Oflfe 






^ /'2»sin(2m+i; 
h sm — 

«r j , sin ^ J \ -^ I 



2 



=1 p5£(??^V(^+2^dd 

fT J sm ^ -^ ^ ' 

1 p- Bin(2».+l)0 

IT jo sm^ -^ ^ ' 

We have therefore to investigate the value to which integrals of this class tend as m 
tends to infinity. Consider in general the value to which 

/./*«£*!^(.)& 

y sm « 
tends when h^ supposed to be an odd integer, increases without limit. 

First suppose 0<A<5 , and suppose that, for values of z within this range, ^ {z) is 
continuous and positive, and that <^ {£) continually decreases as z increases. 

Let -T- be the greatest multiple of t in A, so -r- < A < r+1 t . 



142 THE PROCESSES OF ANALYSIS. [CHAP. Vn. 

Then 

» fw (n+l)ir f» 



Now write 



k 

nw 
k 


+ ...+ / 


•* 

(r-D- 
k 


k 


Binkz 
ainz 


n» 


f * sin 


\kz 


'it\fl» 





so U»' 



ir 

f * siniy - /nir , \ , , nv 

I . /n7 \ »(,T-^y;^' where y=.--p. 



The integrand in this last integral is clearly positive throughout the range of integration, 
and Uj^ is therefore positive. Moreover, under the suppositions already stated, the 
quantity 



decreases as n increases, and it therefore follows that u^ decreases as n increases. 

Also the well-known theorem of Mean Value shews that tin can be represented in the 
form 



where 



ir 

fk ainh/ . 



and Pn-^f^+^K 



S being some quantity between and r. Clearly y^ is positive, and decreases as n 
increases. 

Now we can write 

Jrw smz ^^ *^ 

k 

where J'=Wo-^i+t«|-Ws+...+(-iy"*Mr-i. 

■ 

Since u^ is always positive, and decreases as n increases, we have 

r-1 
where m is any number less than —^ . 

This gives 

< "OPO- («'l-^l)Pt- ("8- »'4)P4- — -("jm-l - O Plm 

-'l(Pl -P8)-»'8 (Ps-Pi)- — - "jm-lCPlm-i-Ptm). 



83] FOURIER SERIES. 143 

As p^ decreases with increase of n, the terms in the last line are negative, and can 
be removed without affecting the inequality. 

Thus 

•^< ^oPto - ("l - "l) Ps - el's - •'4) P4 - • • • - (•'jm - 1 - I'm.) Ptm 
<''0P0-(»'l-»'8)Plin-(»'8-^)Ptm-...-(''lm-l-Vjm)Pfm 

-(»'l-''8)(Pt-Plm)-(»'3-l'4)(p4-Plm)-...-(»'jm-8-«'jm-8)(p2m-2-p2m). 

The terms in the last line are again negative and can be removed. Thus 

•^< "0 (PO - Pjm) + (''O - "1 + ''8 - • . • + ''Sm) Ptm. 

We also have clearly 

J'>tio-tii+w,- ... - 1^.1, 

or •^>«'oPo-''iPi+''8P2- — -^jm+iPiw+i, 

which in the same way gives 

•^>Pfm (''0- 'l + ^S" ••• ~ "Jm+i)- 

Thus J" is intermediate in value between the quantities 

and Pim(''o-''i+*'2- — "-''8m+i). 

Now let k become infinitely great, and let the quantity m likewise become infinitely 
great, but in such a way that t tends to the limit zero. Then the quantities pQ and pi^i 
tend to the limit ^ (0) ; and the quantity 

(am-f 1) V 

f * sinitv J 
or / -: — '- dv 

Jo smy ^ 

mm+Dir sint . 

or 1 ^ cU 

Jo 7 • * 

•^ it; sin T 

k 
can, since k is infinitely large compared with m, be replaced by 



/ 



(2in+l)»8in t ,, 
* 



and this, when m becomes infinitely great, tends to the limit ^ . 

We see therefore that J is intermediate in value between two quantities, each of which 

tends to the same limit, namely ^ ^ (0). J therefore tends to the limit q<I>(0); and 

therefore /, which difiers from J only by a vanishing integral, likewise tends to the limit 

Q<l>(0) as k becomes infinitely great. This result may be called Dirichle^i lenvma. To 

complete the lemma, however, it will be necessary to shew that it is still true when a 
number of the restrictions imposed on ^ {z) are removed. 

(1) It was assumed that if> (z) was positive and steadily decreasing throughout the 
range. 

(a) Suppose that ^ (z) is constant This does not invalidate any of the preceding 
proof, so the theorem still holds if ^ (1) is constant 



144 THE PBOGBSSBS OF ANALT8I& [CHAP. Yll. 

O) Suppose that ^ (z) is negative, or partly positive and partly n^^ative, but still 
steadily decreasing; then choose a constant o so that c+^ (e) is positive through the range; 
then the theorem applies both to o and to o+^ («) and therefore on subtraction to (i) 
alone. 

(y) Suppose that <f> (z) increases steadily throughout the range. Then the theorem 
is true for { — <^ (z)} and therefore for <^ (z). 

Therefore the theorem is still true if <^ (z) is ftmte^ continuous, and steadily increases 
or decreases throughout the range. 

(2) Instead of taking the integral between and A, take it between g and A, where 
0<g<h^-^. We assume that the value of (t>{z) is only known for values of z from 

^ to A. 

Take a new function ^ («), defined as being equal to <^ (g), a constant, for values of z 
from to ^, said equal to (z) for values of z from gU> L Then the theorem holds for 0^ («). 

Also 

^* /la* <^i(«)<&-|*i(0)=|<^(!7), 



and 



Therefore 



by subtraction. 






(3) Now assume there are a limited number n of maxima and minima within the 
range to A. 

Let them be at the values Oi, a^, ... 0^, of s. Then 






On applying the theorem to each of these integrals in succession, it is clear that the 
theorem holds for the whole integral. 

Therefore the theorem is still true if <f) {z) is finite, continuous, and has not more than 
a limited number of maxima and minima within the range. 

It must be noted that these conditions still exclude such functions as e.g. (^-c) sin — 

z^ c 

where 0<c<A. 

IT 

(4) We shall now no longer restrict A to be less than 5- . Take < A ^ ir. 

Then(a)let^<A<tr. 

WriteA=tr-A', where < A' < | . Then 

Lamit /=* / -: <p («) dz + I —. — <f> (z) dz, 

j^moo J sm z J ^ SID z 



83] FOURIER SERIES. 145 

Writing z^n-Cin the latter integral, we have 

ir » 

Limit /« P2iji^*Wcir+ P«^!^*;0(.-f)eif. 

Since <l>{fr-() satisfies the conditions stated, we see that when A' > the second integral 
is zero. 

^ Therefore Limit /- ^ <^ (0). 

O) Let As TT. Then all the above reasoning applies, except that now A' 1*0, so 

Limit/«|^(0)+f<^(ir), 

which, in order to guard against uncertainty in the case in which the function is 
discontinuous at and ir, is often written 

where ff is a vanishing positive quantity. 

(5) Next, suppose that the function <f) (z) within the range has a finite number of dis- 
continuities, in the form of abrupt but finite changes of value. Divide the range into 
various portions, so that each of them ends at one discontinuity and begins at the next, 
and divide each of these into others each beginning and ending at a point of stationary 
value. The above theorems apply to each of the portions, and therefore each integral is 

zero except the first, which is equal to o^(<)) ^^d possibly the last, which when 
h^n has the value o^Ctt-c). 

(6) Finally, consider a function <t>{z) which becomes infinite for z^se, but in such 
a way that the value of / ^ (z) dz tends to a definite limit as z approaches c from either 
lower or greater values. 

Then 

where c is a small positive quantity. 

In the second integral, a quantity ( can be chosen intermediate between c and c- c, such 

sin JkC f^ 
that the integral is equal to -^— ^ / ^ («) c& ; on taking e small this vanishes ; and 

sin C y c - • 

similarly the third integral is zero. 

On making k infinitely large, the fourth integral tends to zero. Therefore the theorem 
holds in this case also. 

(7) Thus we have, summarising the results obtained, the theorem that the Umit when 

jk tends to infinity of I ^^ <l>{z)dzi8^<l){€)ifO<h<Viandis 

Jo sm z 2 

W. A. 10 



146 THE PROCESSES OF ANALYSIS. [CHAP. Vll. 

if h^it ; xohere € is a vanvUhing positive quantity ; provided thai <ft{z) is every where finite, 
and has only a limited number of finite diecontinuitiee and maxima and minima between 
the values and h of the variable z ; and this is still true if <f> {t) has a limited number of 

singularities of specified type, namely such that I <f>{z)dz is finite. 

This result may be called Dirichlefs lemma, the conditions just stated being referred 
to as Dirichlefs conditions. 

We can now return to the expansion which was found to represent the sum of the first 
(2m + 1) terms of the Fourier series. 

We had iS'w=/i+/„ 

X. r 1 /*'''«sin(2wi + l)tf ., , ^A^ jA 

where /.--j^ ^__i_y(,+2tf)rf^, 



1 r*«n(2^) 
^ n Jo am tf -^ ^ ^ 



If 0<;r <29r, and/(j;) satisfies Dirichlet's conditions, we have hj Dirichlet's lemma 

Limit /i— i/(«+€), 

and 

Limit /a=i/(2-€), 

and so 

Limit 5„-i {/(^+,) +/(«_,)}. 

It z^O, we have 

Limit /j = i {/(f) +/(27r - c)}, Limit 7,-0, 



and so 

Limit &,=:i {/(c) +/(27r - f )}. 



If «=2ir, we have 

Limit /i=0, Limit /,-*{/(«) +/(2ir-€)}, 

and so 

Limit 5„=i{/(«)+/(2,r-€)}. 

Thus we finally arrive at Fourier's theorem, namely that the sum to infinity of the 
series 

cio+ S (a^cos9iu+6mSinnu) 

is f{z) at points z for which f is continuous, and is the arithmetic mean of the two values 
off{z) at points z for which f is discontinuous : it being assumed that f{z) satisfies Diriohlet's 
conditions. 

Example, Prove that in the limit when n becomes infinitely great 

/'•8in(2n + l)0 .^,. , ... 

a being a real positive constant. 

(Cambridge Mathematical Tripos, Part IL, 1894.) 



84] FOURIER SERIES. 147 

84 Nature of the convergence of a Fourier series. 

The proofs of Fourier's theorem which have been given establish the result 
only for the case in which the sequence of the terms in the series 

2 (dm cos mz + bm sin mz) 

is that in which m takes the orderly succession of values 1, 2, 3, 4, ... . 

The question now arises whether the order of succession of the terms can 
be deranged without affecting the value of the sum of the series ; in other 
words, we have proved that the expansion of a function by Fourier's theorem 
is a convergent series : we want to find whether it is absolutely convergent, or 
only semi-convergent. The question has also to be considered whether the 
series is uniformly convergent or non-unifomdy convergent in the neighbour- 
hood of a given value of z. 

We shall first shew by considering special cases that there is no geneml 
answer to these questions. 

Consider the series 

sin i^— -sin 2^ + ssin3-«f— ... , 

which represents - z when Okzktt, and ^z^tr when ttkz <2Tr; this series 

is semi-convergent for all real values of z, since sin n.^ is finite for all values 
of n when z is real, and so the modulus of the general term bears a finite 
ratio to the general term of the divergent series 

In this series, therefore, the value of the sum will be modified if the order of 
succession of the terms is changed. 

Moreover, we can shew that the series is non-uniformly convergent at its 
discontinuity tt. For the sum of the first n terms is 

sin 2z . (- 1)***"* sin nz 

smz 5 I-...H , 

2 n 

or 

/ (cos e - cos 2e + ...+(-!)'*"* cos 7rf)cie, 
Jo 

[* fl (- !)**-» co8 (n + l)^-hcosnel , 

Jo [2 "^ ~~2 nr^^^t 'J ^^- 

The term I ^ dz represents the sum of the whole series ; so the remainder 
Jo 2 

after n terms, when — tt < z < tt, is 



or 



cosf»+ gj t 

iin«(-ir'| ^ ^ dt. 

2C08^ 



- 1)"-' 

io 



10—2 



148 THE PROCESSES OP ANALYSia [CHAP. VII. 

Writing <r = 7r — i;, ^■btt — w, this can be written 



i2n=-| 



sin (n+ gj u 

dtL 
28in^ 



Write (n + i j u « I'. The equation becomes 



Sin V 



dv. 



However great n may be taken, if 17 be taken so small that (^ + 5)^ is 

infinitesimal, this int^;ral tends to — I or — -5 , and so is not infini- 
tesimal It follows that the series is non-uniformly convergent in the vicinity 

of -^ — TT. 

Consider next the series 

1 0.1 

COS<r + — C08 3ir + r:C0s5<r + .,. , 
o' o' 

which represents ^^ q ^ when Okzktt, and — — ^ when 7r<z<27r, 

This series is absolutely convergent for all real values of 5, since the moduli 
of its terms are less than the corresponding terms of the convergent series 

1 1 

1 + g, + gj + . . . . 

In this series therefore the order of succession of the terms can be changed 
in any way, without altering the value of the sum of the series ; and since 
the comparison series is independent of z, the series is also tmi/orrnlj/ 
convergent for all real values of z. 

Returning now to the general Fourier series, we can discover the nature 
of the convergence by a. consideration of the coeflBcients in the series, which 
can be made in the following way. 

We have shewn that if 



ao 



then 



1 f *» 
a,» = - 1 f(t) cos mtdt 



84] FOURIER SERIES. 149 

Suppose that (as in most of the examples we have discussed) the range 
0<z<2Tr can be divided into other ranges, say 0<z<ki, ki<z<k^, ,,,y 
kn<z< 27r, which are such that in each of these smaller ranges f(z) is 
an analytic function of j?,, regular iu the range. (f(z) will not necessarily be 
the same analytic function in the different ranges.) Thus it f(z) has the 
value J? for < J? < TT, and has the value --z for ir <z< 27r, we should have 
w « 1 and ki = tt. Then 

1 r *» ,1 /**« 1 f** 

am=- I /(t)coBmtdt + - f(t)coamtdt-\' ... + - I f(t)co8mt(JU. 

Each of these integrals can then be integrated by parts ; we thus obtain 

[*» 1 J.... sin mf\ , r*« 1 ^ ... sin vitl . 

1 r*« 1 f*« 

/ f'(t)smmtdt I f'(t)sinmtdt^ ... , 

TrmJo ''rmJkf ^ 

or 

where 

il = ~ [sin mJe, {/(k, - 0) -/(A, + 0)} + sin mk, {/(k, - 0) --/(k, + 0)} + ...], 

and where bm is the coefficient of sin mz in the Fourier expansion of/' (z) — 
an expansion which will exist, since/' (z) is a function of the same character 
as f(z)y though the terms of this expansion will not always be the derivates 
of the corresponding terms of the Fourier series {ot/{z). 

Similarly 



6 =:? + ^' 
m m 



where 



-B«--[-/(+0) + cosmA;j{/(*,-0)-/(A^ + 0)} + co8mti{/(ifc,-0) 

TT 

-/(A, + 0)} + ...+/ (27r-0)], 
and where a^' is the coefficient of cos mz in the Fourier expansion o{f'(z). 

In the same way we have 

m m 
where 

il' = ^ [sin 7n*i (/' (jfei - 0) VX*i + 0)} + sin TnJfc, {/' (ifc,- 0) V' (*«+ 0)} 



150 THE PROCESSES OP ANALYSIS. [CHAP. VIL 

and 

, , B^ .dm 

where 

dm" and 6m ' being the coeflficients of cos mz and sin mz respectively in the 
Fourier expansion of/'' {z). 

Thus 

A B! a^ 



// 



, B A' bm" 
6w =» - + — 1 r . 

The conditions for the absolute convergence of the Fourier expansion of 
f{z) are therefore expressed by the equations 

il-0, 5 = 0; 

for if these equations are satisfied, we have 



a„ = .^?l±^'and6, ^'"^ 






m* *" m* 



and the terms of the Fourier series are comparable with those of the con- 
vergent series 

1+1+1+1+ 

Now in order that we may have ^ = 0, B = 0, for all values of m we 
must have 

/(*.-0)=/(A, + 0), 

/(Ar, - 0) =/(*, + 0). 



That is to say, if a Fourier series is absolutely convergent for aM real values 
of z, the fwnction represented by the series has no discontinuities, a/nd has 
the same value at z^O as at z^ 27r. 

If these conditions are satisfied the Fourier series is not only absolvlely, 
but is also vmformly convergent For its coefficients a^ and 6m are in this 

case of the order — ; , and so the series of constants 

^ol + jchl + 16, ! + |a,| + |6, 1 +... 



85] FOURIER SERIES. 151 

converges ; but the moduli of the terms of the Fourier series are less than the 
corresponding terms of this series, and consequently the Fourier series is 
uniformly convergent for all real values of z. 

Example 1. Shew that in general, when the Fourier series converges only for real 
values of t^ the quantities a^ and h^^ can be expanded in infinite series of the form 



of which the terms 



m w' m' tn* 



A B .B A' 
5 and - + — , 

found above are the initial terms ; but that when the Fourier series converges within a belt 
of finite breadth in the «-plane, all the coefficients c^) c,, C3, ... vanish, and this expansion 
becomes illusory. 

Example 2. Let /(«) be a function of «, which is regular for all real values of z 
between ««=0 and «=7r, and which is zero at «=0 and z^rr. Prove that if /(«) is 
expanded in a sine series, valid between z=0 and z^fry the series will be absolutely and 
uniformly convergent for all real values of z. 

Example 3. f{z) is a function of z which is regular for all real values of z between 
and TT. Prove that if it is expanded in a cosine series, valid between zs=:0 and z^n, the 
series will be absolutely and uniformly convergent for all real values of z, 

86. Determination of points of discontinuity. 

The expressions for dm, and 6^ which haye been found in the last 
paragraph can be applied to determine the points at which the sum of a 
given Fourier series is discontinuous. This can best be shewn by an 
example. 

Example, Let it be required to determine the places at which the sum of the series 

sin £+} sin 32+i sin 5^+ .•• 



is discontinuous. 




For this series we have 




1 


«m-0, 




, 1 - COS mit 
*»'- 2m" 



Comparing this with the formula found in the last paragraph, we have 

^ = 0, 5=J-j006m7r, 

Hence \i k^^ k^,,, are the places at which the analytic character of the sum is broken, 
we have 

0=^-1 [8mm^i{/(iti-0)-/(iti+0)}+sin mi-, {/(it,-0)-/(it,+0)H...]. 
Since this is true for all values of m, the quantities i{-|, it,, ... must be multiples of n ; but 



162 THE PROCESSES OP ANALYSIS. [CHAP. VII. 

there is only one multiple of n in the range 0<g<2ir, namely n- itself. So k^^v, and 
^2) ^8> ••• ^o not exist Substituting /r|«s7r in the equation ^-■^— ^oosmfr, we have 

J-icoswir- [-/(+0)+COSW7r{/(7r-0)-/(»r+0)}+/(2»r-0)]. 

Since this is true for all values of m, we have 

i=-^{/(2ir-0)-/(+0)}, 
and -j=-l{/(^-0)-/(ir+0)}. 

IT 

This shews that there is a discontinuity at the point x^nt such that 

/(t-o)-/(«-+o)-|, , 

and that 

/(2,-0)-/(+0)=-|. 

Example. Find the discontinuities in value of the siun of the series 

8ini-^sin22+^sin47-}sin5«+^sin7«-|sin82+^sin I0z+,.., 

86. The uniqueneaa of the Fourier expansion. 

We have seen that it / (z) is a quantity depending on z, and satisfying 
certain conditions as to finiteness, etc., then the series 

00+ 2 (Oro COS m« + 6,n sin m;?), 

1 r*' 
where a^ « - I f(t) cos mt dt (m > 1), 



1 r*' 
6m=»- / f(t)s]Ximtdt, 







'^-llj^'^^'- 



has the sum/(^) when ^ -? ^ 27r, except at the isolated points at which /(z) 
is discontinuous. 

The question arises whether any other expansion 

00 

Co + 2 (Cm COS wu + dm sin mz) 

of the same form exists, which also represents f{z) in the interval from 
to 27r ; in other words, whether the Fourier expansion is uniqtte. 

We may observe that it is certainly possible to have other trigonometrical expansions 
of (say) the form 



00+ 2 



(omCOSy+ftnCOSyj 



86] FOURIER SERIES. 153 

which represent f{z) between and %ir\ for write ;?«2^, and oonsider a function ^(C)) 
which is such that <^(f)«/(2f) when 0<f<ir, and <^(0-fl^(f) when 7r<f<27r, 
where g (() is anj other function. Then on expanding ^ (^) in a Fourier expansion of 
the form 

00+ 2 (o^^cosmf+ftncosmf), 

this expansion represents /(«) when 0<;r < 2fr ; and clearly by choosing the function g (() 
in di£ferent ways an infinite number of such expansions can be obtained. 

The question now at issue is, whether other series proceeding in sines and cosines of 
integral multiples of z exist, which differ from Fourier's expansion and yet represent f{z) 
between and 29r. 

If it were possible to have a distinct expansion 

. 00 

/(^) = c^-f 2 (c» COS m* + cijn sin wiir), 

m-l 

then on subtracting this iix>m the Fourier expansion we should have an 
expansion 

(ao-Co)+ ^ {(a« - Cm) cos T/MT + (6,» - cim) sin 7?Mr} 

whose sum is zero for all values of z between and 2ir, except possibly 
a certain finite number of values (namely the discontinuities). 

The investigation therefore turns on the question whether it is possible 
for such an expansion as this last to exist. We shall shew that it cannot 
exist, and that consequently the Fourier expansion is unique*. 

Let' -Ao^gOo, 

-^m =■ c^m cos m-? + 6m sin mr (m> 1); 
and let 

2 = -4.0 + Ai + ... + ilm + ••• 

be a convergent (not necessarily absolutely convergent) series for values of z 
from to 27r, so that the limit of a^ and hn is zero forn:* oo ; and suppose 
that (except at certain exceptional points) its sum is zero. 

Then the series 

ir ^<r;=sulo^ -ill- -^ -...-— -... 

converges absolutely and uniformly for this range of values of z, as is seen 
by comparing it with the series S — . 

We shall first establish a lemma due to Riemannf , which may be stated 
thus: 

* The proof is due to G. Cantor, Journal fUr Math, Lxzn. 
t Collected Worki, p. 218. 



154 THE PROCESSES OF ANALYSIS. [CHAP. YU. 

The quantity 

jy F(z + 2a) -¥ F (z - 2a) - 2F(z) 

^^ 4? 

tends to the limit /(z) as a tends to zero, if at z the series X converges to the 
sumf(z). 

For the term involving On in J2 is 

— -r^i (^ ^^^ n (ir -I- 2a) + a« cos n (-«? — 2a) - 2a^ cos nz], 

On cos nz sin' na 

or T-z , 

nV 

J . ., , ^, ^ . , . I . bn Bin nz sin* na, 
and similarly the term involving bn is -^ r^ . 

As F(z) converges absolutely, we can rearrange the order of the terms, 
and so can write 

n . . /sinoV . A /sin2a\* 

Now considering the series 1, we can write 

say, where z being given, and any small quantity S being assigned at will, we 
shall have | Cn | < S for values of n > some integer m. 

Now An ■* «n+i — en for all values of n. 
Therefore substituting, we have 

Divide the series on the right-hand side of this equation into three parts, 
for which respectively 

(1) l<n<m, 

(2) m + 1 < n < «, where s is the greatest integer in - , 

(3) « + 1 < 71. 

The first part consists of a finite number of terms, each tending to zero 
as a tends to zero, so the first part is zero. 

Considering next the second part, the quantities are of the 

form where <a<w; this quantity decreases as a increases from 

to TT, so the sum of the moduli of the terms in the second part is less than 

^ f/sinmay .sin^a.*] 

which tends to zero when S tends to zero. 



86] FOURIER SERIES. 155 

Considering next the third part, we can write the nth term in the. form 

Ksinn — lay /sinn— laVl «» /• . 7 • • \ 

or 



sin* w — 1 a P 1 _ ll _ sin 2n — la sin o 
^« a* [—J. n«J *** nV ' 

80, as I €n I < S, its modulus is less than 

Thus the whole sum of the terms in the third part 






5 1 sr_i j_ 1 

8 r (ic 8 8 8 .8 



which is ultimately zero. Therefore the three parts of the infinite series 
in R are all zero ; and thus 22 '^fi^) in the limit ; which establishes Riemann's 
lemma. 

Next, we shall establish another lemma, due to Schwartz *, which may be 
stated as follows : If a and b are two of the exceptional points, so that between 
B^a and z^b the series 2 converges to the sum zero, then F{z) is a linear 
function of z betu)een these values. 

For assume that a is less than b, and introduce a function 4> (z\ defined by 
4,{z)~0^F(z)-F(a)-l-^^{F{b)-F(a)\^-^{s^aHb-z). 

where ^s 1 and h is any constant 

Then substituting in the result of Riemann's lemma, we have 

Therefore (^ + a) + ^ (^ - a) — 2^ (z) is positive when a is very small, 
whatever be the value of z. 

Now ^(a)«0 and ^(6) = 0. Also ^(z) is continuous, since F(z) is 
uniformly convergent, and consequently continuous. Therefore if <f> (z) can 
be positive between the values a and b of z, it will have a maximum ; 
let this occur at the value c of z, 

* Quoted by G. Cantor, Joumaifiir Math. Lxxn. 



156 THE PROCESSES OP ANALYSIS. [CHAP. VII. 

Then when a is small, we have 

<t>(c + a)-4>(c)<0, and <^(c-o)- <^(c)<0. 

Adding these relations, we see that the condition just found is violated, and 
so <f) {z) can not be positive at all within the range. 

Again, take h small. Choose ^ » j: 1, so choosing the sign that the first 
term ^ [i^ (x:) — . . .] is positive. Then ^ {z) is clearly positive, if this first term 
is not zero. 

But j> {z) is not positive ; and thus we must have 



Therefore F{z) ia ^ linear function of z, which establishes Schwartz's 
lemma. 

We see then that the curve y = F(z) represents a series of straight lines, 
the beginning and end of each line corresponding to an exceptional point ; 
and as F{z), being uniformly convergent, is a continuous function of z, these 
lines must form parts of a polygon. 

But by Riemann's lemma 

limit ^i' + '0-f(') F(z-a)-F(z) ^Q 

Now the first of these fractions gives the inclination of the earlier side of 
the polygon at a vertex and the second of the later ; therefore the two sides 
are continuous in direction, so the equation y=^F(z) represents a single line. 
If then we write F (z) ^cz-^ c\ it follows that c and c' have the same values 
throughout the range. Thus 



and therefore 



A ^ A ^tl I 






the right-hand side of this equation being periodic, with period 27r. 

The left-hand side of this equation must therefore be periodic, with period 
27r. Thus we have 

4^ = 0, c = 0, 

n h 

and — c's=ili-H...-H-r cosn^4-~sinn5-H.... 

nr n' 

Now the right-hand side of this equation converges uniformly, so we can 



86] FOURIER SERIES. 157 

multiply the equation by cos nz, sin nz, respectively, and integrate. This 
gives 

TT ~ =« - c' I COS nz4^ « 0, 
^ J 

b [^ 

and TT -4 « — c' I sin nzdz « 0. 

n« Jo 

Therefore the a's and b's vanish, so all the coefficients in S vanish ; which 
establishes the result that the Fourier expansion is unique. 



Miscellaneous Examples. 

m 

1. Obtain the expansioiis 

(«) 1 — o — ^=H-rooe*+r«coe2f+... , 

^ * l-Srcoe^+r" 

W 5log(l-2rooe«+r^— -rcoB«-jr*co8 2«-5r"ooe3;«-..., 

(c) tan"** _ — r8in«+5r*8in2r+5r"8in3;«+..., 

^ * l-rcoB« 8 3 ' 

(a) tan"*-= z-=r8m«+s*'^sui3«+rr*8m5*+..., 

^ ' 1—1* 3 5 ' 

and shew that, when | r | < 1, they are oonvergent for all values of t in certain belts 
parallel to the real axis in the ^plane. 

2. Shew that the series 

- sin — sin^ 8in(n+l) — sin | 

where all the terms for which i; is a multiple of n are omitted, represents the greatest 
integer contained in a, for all real values of z between and n. 



3. Shew that the expansions 

^l0g(2<X)8|j 



cost - 5C0s2;;+xC0S32... 



and 



- log(2 8in rj= -co8«-5 008 2;j-s008 3«... 

are valid for all real values of «, except mtdtiples of ir. 
4. Obtain the expansion 

• (-l)*^co8fn« , «., /„ '\ . ' / • o . • \ 

and find the range of values of t for which it is applicable. 

(Trinity College, 1898.) 



158 



THE PROCESSES OF ANALYSIS. 



[chap. VII. 



5. Let n be an integer ^ 2, and let XnX^^... x^^i be quantities satisfying the conditions 

0<a?i<ar,<...<ar,i_j<l, 

and write Xq=0, ^n**!* 

Let 0^, 0], c^t ... c^.i be real arbitrary constants and let a function <f> {a) be defined by 
the equalities 

<^(ar)=c^+(Ji+..,+c„ for ar,<a?<it:,+i (i—O, 1, 2, ...n-l), 

<^(x)-Coforx«Xo, 

Cm 



<^(^)-Co+C|+...+c^i+^, for x^Sf 



Shew that 



for X^Xn* 



(*-l, 2, ... n-1), 



2 Mai 



for 0<a?<l, 



and 



*jWjt(i2.5+ i «.. 



where the coefficients a^ and 6^,^ are given by 

11-1 
ao«2 2 (v(l-.JPr), 



o,^" 2 (V8in2m9r^r> 

wiir 



for m'^ly 



1 »-* 
6«=--- 2 (v(l-coe2mira?r) 



for m^l. 

(Beiger.) 

6. Shew that between the values - v and + ir of t the following expansions hold : 

3 sin 3s 



mximz 



2 . / sini 2 sin 2s 3 sin 3s \ 



2 . 
COS msB- sin imr 



(_1_ rncos* fnoos2s moos3s \ 
2^'^F^^*'' 2«-m« ^ 3«-m« " "7' 






-Hg"*^ ^2 / 1 moosg 



m cos 2s wicos3 s 
2*+«? " 3*+wi 






7. Obtain the expansions 



5 sin/i(f-fmtr) 
■-00 s+m*r 



sin(2n-|-l)s 



and 



• cos/*(s+mfr) 

2 



IB — 00 



s+mir 



Sin I 
,sin2nscoti 

' oos(2n + l) s 
sins 

cos 2ns cot z 



(2n</i<2n+2) 



0A-2nX 



(2n</i<n+2) 



0»-2nX 



MISC. EXS.] FOURIKB SERIES. 159 

lip and q are positive integers, shew that 

^ Bin (am -k-p) — ^ 

2 i I »~8m — ^- cot^--, 

».— ym+jt) ^ ^ ^ ' 

cos(^+p)?^ 
* ^^ '^ a IT 2npir .^pn 

2 * aa — cos ^ COt-^— . . 

m— «D ^-m+jt) ^ ^ q 

8. Prove that the locus represented by 

2 - — I — sinitrsinnyaO 

is two systems of lines at right angles, dividing the coordinate plane into squares of 
area nK 

(Cambridge Mathematical Tripos, Part I., 1895.) 

9. If m is an integer, shew that 

^ o 1.3.5...(2m-l) fl , »i o . m(m-l) . 

2. 4. 6... 2m (2 m + 1 (fn + l)(m+2) 

m(m-l)(m-2) 1 

■*'(m+l)(i»+2)(nH-3)*^'^'^-7 
(a terminating series), 

^, 4 2.4.6...(2m-2) fl ^2i»-l ^ r2m-l)(2m-3) ^ . ) 
^^'^^■^ 1.3.5...(2m-l) i2-^2iM:i^^-^ (2m-hl^^ 

(an infinite series). 

Shew also that 

, 4/ oos3s.co8 5;s oos7«.oos9« \ 



and 

cos 



1 - / oosg , ooe3g ^ oosSg cos7i co6 9f \ 

' *"irV1.3"^1.3.6 3.5.7"*"6.7.9'"7.9.1l"^*'7' 



10. A point moves in a straight line with a velocity shich is initially ti, and which 
receives constant increments, each equal to ti^ at equal intervals r. Prove that the velocity 
at any time t after the beginning of the motion is 

u ^ ut u • 1 . 2mirt 
- + — +- 2 -am — , 

and that the distance traversed is 

trf ,^ . . . Mr VT • 1 2mirt 
2;('+") + i2-2,ri,.!,JSS^-T-' 

11. Shew that 

• / .« . sinarir • (-l)*sin(a+2nvir) 

sin(a + 2tMr9r)» 2 ^ ^— ^ ', 

It .00 x—n 

where n is the difference between the real quantity v (supposed not to be an odd multiple 
of \) and the integer to which v is most nearly equal 

(Cambridge Mathematical Tripos, Part II., 1896.) 



160 THE PBOCfiSSES OF ANALYSIS. [CHAP. VIL 

12. Let 9t be an integer > 3, and let ^o, ^|, ^2> <•• be an infinite aet of quantities, which 
satisfy the conditions, 

Let X be a real variable, and let s be the greatest integer contained in nx. 
Shew that when x > 0, 

2 ^r= o + 2 {(hnCoa2inirx+b^sin2mnx)f 



if r is not a multiple of - ; 



2 



but 



2 ^r-o "?+ 2 (aMOOs2fnir^H-6m8in2in9r^, 

2 2 masl 



if r is a multiple of - ; the coefficients a^ and h^ being determined by the formulae 

fi 

^" " •n^ 2 ^,.8in -— (m^ 1), 

1^ n— 1 2fitr9r 

^"^ — 2 9r<^—— («i>l). (Berger.) 

13. Let ^ be a real variable between and 1, and let n be an integer > 5, of the form 
4m+ 1, where tn is an int^;er. 

Let E (a) denote the greatest integer contained in a. 
Shew that 

(-1) v»/ + (— 1) v « / = _ + _ 2 —tan cos2m9r^, 

itx\a not a multiple of - ; 



but 



. nnx . ooBnnx 2.2 • 1 ^ 2mfr 
sin ^ + — ' — ■« - + _ 2 — tan cos 2mirx. 



if 0? is a multiple of - . (Berger.) 

14. Let a; be a real variable between and 1, and let n be an odd number ^ 3. 
Shew that 

( — 1)*«- + — 2 —tan — cos29n9rar, 
if 0? is not a multiple of - , where < is the greatest integer contained in nx\ but 

tv 



MISO. £XS.] FOURIER SERIES. 161 



0=-+— 2 -tan — coa2inirx. 
n n si3| m n 



1 



if 4? is a multiple of - . (Berger.) 

15. Let X denote a real variable between and 1, and let 9i be an integer > 3 ; further, 
let E(a) be the greatest integer contained in a. Shew that 

n wi IT m»i f^ n 

if ^ is not a. multiple of - ; but 

. ^1 (n-l)(n-2)^l • 1 .mir ^ 

tu;*— Twr+s" ^ ^ + - 2 — cot - co8 2m»rj?, 

as on . IT mmi m n 

1 

if ;p is a multiple of - . (Berger.) ' 

tit 

16. Assuming the possibility of expanding f(x) in a series of the form lAjg sin kx, 
where 1; is a root of the equation i&co8ai;+6sinai:=0, and the summation is extended 
to all positive roots of this equation, determine the constants Aj^, 

(Cambridge Mathematical Tripos, Part I., 1898.) 



17. If 



shew that 



6*-l ftsO ^J 



. C0s4Yr^ . C0s6irJ7 . , ,v. , 2**~*ir** „ , . 

cos2ir^+— 2^- +— 3s^+... = (-l)"-^-2;jj- V^(x\ 

ainSiixl""^'"^ I »^^^^^ | _/_n«+i?!!!!:^V (x) 



18. If 



shew that 



(Cambridge Mathematical Tripos, Part II., 1896.) 



/(x) = ^Oq + Ox cos a;+ a, cos 2^ + . . . , 



If 
shew that 



a^ss" I f{x)ooanxtsji^x -. 
If J ^ 



<^ (j*)=6| sin ^+62sin 2a?+ ... ) 



b^xM - I <l}{x)axinx tan ^a? — . (Beau.) 



19. Prove that the series 2ii»sin - — , 

1 <* 

where ^n^zl sm -— -/(t;) cfv, 

is equal tof(x) for any value of x lying between and a about which f(x) is continuous. 
W. A. 11 



162 THE PROCESSES OF ANALTSia [CHAP. VIL 

If /(O), /(a) are the limits of /(«), /{a^^tX when the positive quantity c diminishes to 
zero, and if /(:r) has sudden increases of value A, k, corresponding to the values a, /3, ... of x, 
the limit for 7»= ao of nA^ can be written in the form 

i|/(0)-(-l)./(«)+Acoe^+*co8^ + ...}. 

Shew that the series 

sin3a;+- Bin9x+-sin 15:t7+...-2 (sina?+- sin3:F-|--sin 5jf-|-...] 

^3^3/ . 1 . K . 1 • ^ 1 • 11 ^ 1 

H -jsmd?- r|8ln5ar+=jSm7A•-_-T^smll4?+..,V 
ha8 the limit -^n* when Xy lying between and 9r, approaches indefinitely near to one or 
other value, and that it has sudden changes of value -^ir and H-^fr corresponding to the 

values i n and § n- of :r. 

(Cambridge Mathematical Tripos, Part I., 1893.) 

20. If, for all real values of x, 

F{x) = AQ+AiCosx+AiCOB2x+A^co&3a;+,,,, 
then 

cos(^jF(^)d^= / XF(x)dx, 

where 

£r= il^ -I- Jj cos IT + il J cos 4tr + -ij cos 9tt? + . . . , 

F=4i sin i^+^ij sin 4M;+il3 sin 9tr+ ... , 

-, X^ ^^ An^^-X^ ira? , ^ 4(2»r)«+a:» 2ir:r . 

^=COS -;-+2C03 COS h2C0S — — J- COS + ... 

Aw Aw w Aw w 

4(2n»r)«+a:» ^nirx 

+ 2 COS -^ — 7 cos . 

Aw w 

Prove these formulae, and thence deduce the result 

{U+ V) (^y-l^(0)+^Mco8^^'?+/'(2«-)oo8 ?*;"+... 

2ir 
where to^-r- ^ k being a positive integer. When k is even, the last term of each series 

involves F{ikw) and is to be multiplied by ^; when k is uneven, the last term involves 

F{i{k-l)io), 

(Cambridge Mathematical Tripos, Part II., 1896.) 






CHAPTER Vni. 
Asymptotic Expansions. 

87. Simple example of an asymptotic expansion. 

Consider the function 

• el^Ht 



J 3 



t 



where x is real and positive, and the path of integration is the real axis in 
the ^-plane. 

Integrating by parts, we have 

and by repeated integration by parts, we obtain 

^/ \ 1 1.2! . (-!)»-' (»-l)! / ,w , r«^*d« 

In connexion with the function f{x\ we therefore consider the series 

1 1^2! ^ (-!).-.(„ -1)1 

X oc^ a? '*' x^ 

We shall denote this series by S, and shall write 

1 12! (-l)nn! _ 

X ^■*"a;» •••"^ (^"^^ "'^~* 

n-1 

The ratio of the nth term of the series & to the (n — l)th term is : 

^ ' X 

for values of n greater than 1 + a?, this is greater than unity. The series 8 is 

iherefo7'e divergent for aU values of x. In spite of this, however, the series 

can under certain circumstances be used for the calculation of f{x); this can 

be seen in the following way. 

11—2 



164 THE PROCESSES OF ANALYSIS. [CHAP. YIII. 

Take any definite value for the number n, and calculate the value of Sn^ 
We have 



/(^)-Sfn-(-l)«+'(«+l)!/J'^f. 



and therefore 



\/(x)-8n\=(n + lV.( 

J i 






— ^ , since e*^ < 1 and t is positive^ 



< 



^n+i- 



For values of x which are suflBciently large, the right-hand member of 
this equation is very small. Thus if we take x > 2n, we have 

1 



|/(^)-^n|< 



2«+i n 



1 > 



which for large values of w is very small. It follows therefore that the value- 
of the fwaction f(x) can be calcukUed with great accuracy for large values 
of X, by talcing the sum of a finite number of terms of the series 8. 

The series is on this account said to be an asymptotic expansion of the- 
, function f{x). The precise definition of an asymptotic expansion will now^ 
be given. 

88. Definition of an asymptotic expansion, 
A divergent series 

-^0 + — H"-^ +.••• + ^ + •••» 

in which the sum of the (w + 1) first terms is Sn, is said to be an asymptotic 
expansion of a function f{x), if the expression af^ \f{^) - Sn] tends to zero 
as X (supposed for the present to be real and positive) increases indefinitely- 
When this is the case, if a? is sufficiently great, we have 

where € is very small ; and the error — committed in taking for f{x) the- 

(n-f 1) first terms of the series is very small. This error is in fact infini- 
tesimal compared with the error committed in taking for f{x) the n first, 
terms of the series : for this latter error is 

of" ' 
and € is in general infinitely small compared with iln + €. 



88,89] 



ASYMPTOTIC EXPANSIONS. 



165 



The definition which has just been given is due to Poincar^*. Special 
asymptotic expansions had, however, been discovered and used in the 
eighteenth century by Stirling, Maclaurin and Euler. Asymptotic expan- 
sions are of great importance in the theory of Linear Differential Equations, 
and in Dynamical Astronomy ; these applications are, however, outside the 
scope of the present work, and for them reference may be made to Schle- 
singer 8 Handbuch der Theorie der linearen Differentialgteichungen, and the 
second volume of Poiucar6's Les M4thode8 Nouvdles de la Micanique^ Cileste. 

The example discussed in the preceding article clearly satisfies the 
definition just given : for 

n\ 



and the right-hand member of this equation tends to zero as x tends to 
infinity. 

The term "asymptotic expansion" is sometimes used in a somewhat 
wider sense ; if F, ^, and / are three functions of a?, and if a series 

X or 
is the as3anptotic expansion of the function 



F ' 



we can say that the series 



^ „. FA^ FA^ 



X 



as" 



is an asymptotic expansion of the function J, 

For the sake of simplicity, we shall consider asymptotic expansions only in connexion 
with real positive values of the argument. The theory for complex values of the argiunent 
may be discussed by an extension of the analysis. 



89. Another example of an asymptotic expansion. 

As a second example, consider the function f{x), represented by the 
series 






.(1), 



where c is a positive constant less than unity. 

The ratio of the Arth terra of this series to the {k — l)th is less than 
unity when k is large, except when a? is a negative integer, and conse- 

♦ Acta Mathematical vin. (1886), pp. 295—344. 



166 THE PROCESSES OF ANALYSIS. [CHAP. VIII. 

quently the series converges for all values of x except negative integral 
values. We shall confine our attention to positive values of x. We have, 
when x>k, 

1 I k k" 1^ h 

= = + -^--.+ -7-.... 



X + k X (/^ s? SK^ a^ 
If, therefore, it were allowable to expand each fraction , in this way, 

X T" fu 

and to rearrange the series (1) according to descending powers of a?, we 
should obtain the series / 

t4"-^-4'^ w- 



00 



where ili= 2 c*; il3 = — 2 A;c*, etc. 

But this procedure is not legitimate, and in fact the series (2) diverges. 
We can, however, shew that the series (2) is an asymptotic expansion of 
/(a?), which will enable us to calculate /(a?) for large values of x. 

Forlet' s„ = =^ + ^» + ...+^'. 

X a^ af^^^ 

and ^{/(^)-£f„} = LjL2_ 2 ^. 

Now 2 r is finite, and so when x is infinitely great the right-hand 

t— 1 X -h K 

member is infinitesimal. 

Therefore a?"{/(a?) — /Sn} tends to zero when x tends to infinity; and so 
the series (2) is an asymptotic expansion of f(x). 

Example. l{f(x)= I e^^^dt^ where x is supposed to be real and positive and the 
path of integration is real, prove that the divergent series 

\ 1 1.3 1.3.5 

is the asymptotic expansion otf(x). 



90] ASYMPTOTIC EXPANSIONS. 167 

90. Multiplication of asymptotic expansions. 

We shall now shew that two asymptotic expansions can be multiplied 
together in the same way as ordinary sehes, the result being a new 
asymptotic expansion. 



For suppose that 






X a^ '" af^ 



•^0 '• "T" "^ ^^"T •••4" _„■ > ••• 



are asymptotic expansions representing functions J{x) and J'{x) respectively, 
and let 8^ and 8n be the sums of their (n + 1) first terms ; so that 



Limit a?« (/ - 8n) = 0^ 

(1). 

Limita;»(J-'-/S„') = 

ap=300 , 

Form the product of the two series in the ordinary way ; let it be 

and let Sn be the sum of its n first terms. 

As 8n, 8n and 2» are simply polynomials in - , we have clearly 

X 

Limit «»(flf„S„'-S„) = (2). 



gmto 



Now by (1), we can write 



'^ = ^« + ^' 






where Limit e = 0, Limit e' = 0. 



ee' 



Then a!» (J J' - S„ 5«') = Sn'e + /S„€' + ^: . 

X 

The terms in the right-hand member tend to zero as x tends to infinity. 
Hence 

Limit a?«(JJ'-Sn/Sfn')=0..... (3). 

«B00 



168 THE PROCESSES OF ANALYSIS. [CHAP. VIIL 

From (2) and (3) we have 

Limit a^ {J J' - 2n) = 0, 

X=oo 

and therefore the series 

X or 
is the asymptotic expansion of the function JJ\ 

91. Integration of asymptotic expansions. 

We shall now shew that it is permissible to integrate an asymptotic 
expansion term by term, the resulting series being the asymptotic expansion 
of the function represented by the original series. 

For let the series 
represent the function J(x) asymptotically, and let Sn denote the sum 

Then, however small a real positive constant quantity € may be taken, it is 
possible to choose x so large that 



and therefore 



J(<')-Sn\<^, 



\r J{x)dx-{ Sndx\^r\J{x)-8n\dx 

\j 9 J X I J » 



^(n-l)«»-»' 
and therefore the integrated series 

a, ^2<r»+-+(n-l)«»-> + - 
is the asymptotic expansion of the function 

J{x) dx. 



J a 



X 

On the other hand, it is not in general permissible to di£ferentiate an asymptotic 
expansion. 

92. Uniqueness of an asymptotic expansion. 

A question naturally suggests itself, as to whether a given series can be 
the asymptotic expansion of several distinct functions. The answer to this 



91, 92] ASYMPTOTIC EXPANSIONa 169 

is in the affirmative. To shew this, we first observe that there are 
functions L{x) which are represented asymptotically by a series all of 
whose terms are zero, i.e. functions such that 

Limit x^L {x) = 0, 

ivhatever n may be, when a; (supposed to be real and positive) increases 
indefinitely. The function e~* is in fact such a function. The asymptotic 
•expansion of a function J{x) is therefore also the a83anptotic expansion of 

J{x)-¥L{x). 

On the other hand, a function cannot be represented by more than one 
-distinct asymptotic expansion for real positive values of x ; for if 



X a^ 



-4o+— + —'+.. . 



-and jBo + — +-?+... 



X a^ 



«re two asymptotic expansions of the same function, then 

U^i^(A4" + ...4--A-4....-|.).„, 

which can only be if ul© = 5© ; -^i » -Bi, etc. 

Important examples of asymptotic expansions will be discussed later, in connexion 
with the Gkunma-function and the Bessel functions. 



Miscellaneous Examples. 



1. Shew that the series 



1 1 ' 2 ' 3 ' 
- + —+ — + — -I- 



is the asymptotic expansion of the function 



/ 



dt 

t 



when X is real and positive. 

2. Discuss the representation of the function 

(where x is supposed real and positive, and <^ is an arbitrary function of its argument) by 
means of the series 



170 THE PROCESSES OF ANALYSIS [CHAP. VDI. 

Shew that in certain cases (e.g. (f)(t)^e^ the series is absolutely convergent, and 
represents f{x) for large positive values of j? ; but that in certain other cases the series is 
the asymptotic expansion o{f(x). 

3. Shew that the divergent series 

I a-l (a -l)(a-2) 

is the asymptotic ezpemsion of the function 

«-• r* 

log^j, 
for large positive values of z, 

4. Shew that the function 



/(.)=/; (iog«-.iog(^)}^- 



6-»» 



has the asymptotic expansion " 

where By^B^y ... are Bemoulirs numbers. 

Shew also that/(j;) can be developed as an absolutely convergent series of the form 

5. Shew that the function 

has the asymptotic expansion 

1.3.*.(2n-3) 



00 



PAET II. 



THE TRANSCENDENTAL FUNCTIONS. 



CHAPTER IX. 



The gamma-function. 



93. Definition of the Oamma-function : Euler*8 form. 
Consider tlie infinite product 



-n 

n 



This product clearly diverges if -e is a negative integer, for then one of 
the denominator-factors vanishes. If z is not a negative integer, the product 
will (§ 23) be absolutely convergent, provided the series 



Jj.log(l + l)-log(l + i). 
is absolutely convergent ; but since when n is large we have 

the terras of this series ultimately bear a finite value to the terms of the 
series 



and therefore to the terms of the series 2 — , which is absolutely convergent. 

The infinite product is therefore absolutely convergent for all values of z, 
except negative integral values. 



174 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 

This product may be regarded as the definition of a new function of the 
variable z\ we shall call it the Oammorfunction, and denote it by r(^), 
so that 



r{z)=-u 



i.('-S" 



n 



This form of the function was fii-st given by Euler ; but the notation r(^) 
is due to Legendre, who applied it in 1814 to an integral which will presently 
be discussed, and which represents the Gamma-function in some case& 



Excunple. Prove that 



r(;?)= Limit -r-r -ryr/— — ^T^'• (Gauss.) 

•«=«, «(«+I)...(«+n-l) 



94. The Weierstrassian form for the Oamrnxi-fmiction. 

Another form of the Gamma-function can be obtained as follows: 

We have 

, n+l 

v{z)^- n^ 

*»-! 1 + 1 

n 
= iLimit «*><«'"'+« fi ^ 

Z fHsoo 



"-('^9 



= - Limit 6'{"*<~+"-'-«---»}n -21-. 



1+- 

n 



Now ! + _+.. .+--.-log(m+l)= t (--log 



n+1 



n 



= 1 ni-±-)i^ 

Joln-iw(n-fa?)j 



Now the series 2 —. r is absolutely and uniformly convergent for 

real values of a? between and 1, as is seen by comparing it with the series 

* 1 

n=l ^ 



94] THE GAMMA-FUNCTION. 175 

hence as m increases, the right-hand member of this equation tends to the 
limit 

1 ( 00 



io(n=iw(n+"^ 



which is finite, since the range of integration is finite and the sum of the 



w 



series 2 —7 r is finite. This limit is known as Euler's constant, and we 

ii=in(n+a?) 

shall denote it by 7, Its numerical value is 

0-5772157. ... 

Thus Limit ]i + h+... + log(m + l))- = 7, 



z 



1 * 6 

and 80 r (^) = - e-^' II 



or 



^ n=i 2 , f^ ' 

n 

}-^zey'U |fl + ^)^"*l 

W n=l (V n) J 



This form (due to Weierstrass) shews that yr7-\ is a regular function of z 
for all values of z. 

Example 1. Prove that 

1^(1)= -y, 

where y is Euler's constant. 

For differentiating logarithmically the equation 

and putting z=\ after the differentiations have been performed, we have 

-r'(i)=i+y+i(^^-l). 

or r'(l)«-y. 

Example 2. Shew that 

2 3 n Jo » 

and hence that Euler's constant y is given by 

' Jo « 

Example 3. Shew that the infinite product 



176 TBANSCENDENTAL FUNCTIONS. [CHAP. IX, 

has the value 



For nil — )««= n e* 

i»»i \ z-¥nJ n=i n+z 



*^* {n+z)e'H 



5 (i+izf) 



e n 



n fl+-^tf-^ 

The numerator of this expression is Weierstrass' form of 

1 

(«-j?)6>^-'>r(a-a?)' 

and the denominator is 

1 



ze^'riz) 
Therefore the given expression has the value 

e^zr (z) 



96. The difference'equaiion satisfied by the Gamma-function, 

We shall now shew that the function r(^) satisfies the difference- 
equation 

r(^ + i) = ^r(^). 

We have 

1 «(^ + 9 

r(^+i) = -^n — 2i_ 



1 + 

n 



IVn-^l 






z-^-ln^i n-^^ + l 



n 



00 



1 + 



71/ il V W/ 



u- — —= n 



n+1 n 

This is one of the most characteristic properties of the Gamma-function. 
It follows that if 2r is a positive integer, we have 

r(z) = (^-i)! 



96,96] THE GAMMA-FUNCTION. 177 

Example, Prove that 



111' 



e 



r(r+i) r(z+2) ' r(«+3) 

\z l!«+l'*"2!«+2 Sl^+a"*" •••J • 
For consider the quantity 

1 1 1 

^'*"«(i5+l)'*"^(«+l)(z+2) **■•••• 

This can be expressed as the sum of a number of partial fractions, in the form 

z ^«+l^**'^«+n^*'" 
To find the coefficients a, multiply by {z-\-n) and put «= -n ; we thus obtain 

1 fll 1 )(")*« ^ 

^"'(-i)"7i!r'*'i'*'r2"*"r70"^ "J "~w^ * 

Therefore 



z 
But 



2^«(2+l)^z(«+l)(z+2)^- ^\2 (0+1)1! (^+2) 2! -J 



1 r(«+n+l) 



«(z+i)...(«+«) r(2) 

whence the required result follows. 

96. Evaluation of a general cla^s of infinite products. 

By means of the Gamma-function, it is possible to evaluate the general 
class of infinite products of the form 

00 

n Un, 

where v^ is any rational function of its index n. 

For resolving Un into its factors with respect to w, we can write the infinite 
product in the form 

Yi j Q/t - ai)(n- g,) ... (n-aj 
„«i ( (n-6i) ...(n-6j) 

In order that this product may converge, it is clearly necessary that the 
number of factors in the numerator may be the same as the number of 
factors in the denominator; for otherwise the general term of the product 
would not tend to the value unity as n tends to infinity. 

w. A, 12 



178 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 

We have therefore A: = /, and can write (denoting the product by P) 

n-il(n-6i) ... (n-6fc)j 
For large values of n, this general term can be expanded in the form 

('-3-('-?)('-r-('-r' 

or 1 h terms m — r + . . . . 

n n' 

In order that the infinite product may be absolutely convergent, it is 
therefore further necessary that 

Oi + ... + ajb — 6i — . . . — 6t = 0. 

We can therefore introduce a fiwtor 

e ^ 

in the general term of the product, without altering its value ; and we thus 
have 

But n j(l--)e"U „/ - .^^ . 

Therefore P= ^^^^dhlM (=*«) •_-.l»£(zi_M 

a formula which expresses the general infinite product P in terms of the 
Gamma-function. 

Example 1. Prove that 

•J" «(«_-H5>+f) ^r(a+i)^r(6+i) 

,=i<a+«)(6 + *) "r(a + 6 + l) * 
Example 2. Shew that 

^(i-g)^i-g)...-{-r(--:»;^)r(-a^)...r(-a-iJ)}-S 

where 

2ir . . 2ir 

a=cos hi Sin — . 

n n 



\ 



97, 98] 



THE GAMMA-rUNCrriON. 



179 



97. Conneodon between the Gammorfunction and the circular functiovs. 

We now proceed to establish another of the characteristic properties of 
the Gamma-function, expressed by the equation 



r(z)r(l-z)=^ 



IT 



SUITTZ 



We have 



2(i—Z) „.i 



(•^3 



]^\«+l— « 



W-f 1 



1 • 

n 



1 • 
n 



Zil-Z) n^i 



=i n ^ 



(•-9('-.-Tl) 



'■(-5) 



TT 



Sin TTZ 



which is the result stated. 

Corollary. If we assign to z the special value g, this formula gives 

|rg)('=..orr(i) = .* 

98. The multiplication-theorem of Oauss and Legendre. 

We shall next obtain the result 

»-i 

r(^)r (^ + 1) r [z+fj ... r (^ +^) = r (n^)(2,r) * «*-». 



n 



ni; 



For let 



0(^) = 



r(^)r(^ + ^)...r(;r+ 



n-l 



w 



Then 



<l>(z 



nF (n^) 

^rur+nf (^ + 1) F (^ + 1 + i) ... F (^+ 1 + ^^) 

nF (n-2 + w) 



(m + n - 1) (nz + » — 2) ... (nz) 



^W 



12—2 



180 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 

It follows from this that ^ (2) is a one- valued function of 2, with the period 
unity ; and <f> (z) has no singularities when the real part of ^ is positive, since 

=-7 — r is ever3rwhere regular ; it has therefore no singularity for any value 

of z, and so by Liouville's theorem (§ 47) it is a constant. 

Thus <j) (z) is equal to the value which it has when ^ = - ; which gives 

The«f.„ *F^- {r (1) r (I - 1)) {r (?) r (1 - 1)} 



(by § 97) = 1:11 ^ (2^^!: 



Thus <l>(z) = 



. TT . 27r . (n — l)7r n 

sm - sm ... sin ^ — 

n n n 

«-i 



or 



r(^)r(^+^) ...r(^+'^) = r(n^)n*-~«(27r) 



n-1 
2 



Example. If 
shew that 

Binp,nq)=n-'^ it(g,g)iK2g,g^ ... ^K>^- l)g, g} ' 

99. Expansions for the logarithmic derivates of the Gammorfunction. 
We have 

{r (z + !)}-» = ^^n (1 -I- ^ €~\ 

DiflFerentiating logarithmically, this gives 

dlogr(^ + l) _ f ^^ 1 1 \ 

dz '^■^^(l(^+l)"^"2(^+2)"'"3(^ + 3)"^"y 

Also 

iogr(^ + i) = iog^ + iogr(^), 

so 

^iogr(.+i)-U^iogr(.). 



99, 100] THE GAMMA-FUNCnON. 181 

Therefore 

^iogr(.) = l+*iogr(.+i) 



1 d_( z 



+ 1) "^'2(^ + 2) ■*■••• 

_ 1 1 1 

"■^«'*"(^ + l)>"^(^ + 2)«^-- 

These expansions are occasionally used in applications of the theory. 

100. Heine's expression ofT{z)asa contour integral. 

It has long been recognised that the Oamma-fiinction is intimately 
connected with the theory of a large and important group of definite integrals ; 
and in fact the function has frequently been defined by means of a definite 
integral. We now proceed to consider various definite integrals in this 
connexion, the most general of which is due to Heine and can be obtained 
in the following way. 

We have /, IV 



i^('-3" 



r(^) = - n 

m 

1 ^- 7n 
Now if we express - 11 in partial fractions, we obtain 

i n — = 1 (-i)" — — =— 



^ m»i z-^m mao tn ! (n — m) \ z + m 

Consider now the function 

(- ^)-'. 

This, when a is a complex quantity, may be defined as being equiva- 
lent to 

r«-i)iog(-») 

Now the logarithmic function is many-valued, since the value of the 
function log (— w) is increased or decreased by 27ri when the variable x describes 
a simple circuit round the point a? = 0. In order that the function (- a?)*"* 
may have a unique value, we have therefore to select one of the different 
determinations of log (-a;): and this may be done in the following way. 

We first make the stipulation that the variable x is not to cross the real 
axis at any point on the positive side of the origin ; this prevents x from 
making circuits round the origin, and so makes each of the determinations of 
log (— x) a single-valued function. Then we select, fix>m these determinations, 
that one which makes log (— x) real when ^ is a real negative quantity. The 
value of log (— x) being thus uniquely defined for every value of x, it follows 
that the value of (—a:)*"* is likewise uniquely defined. 



182 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 

With these presuppositions, if (7 be any simple contour enclosing the origin 
and cutting the real axis in the point x^l,we have clearly 

f , , , , r (-a?)»l 6^ c-'*^ 2tsin7ra 

Jc L(7 « J a a a 

Therefore 

iniLl^=,-.i-j(i+i)...(i+i)ri f (-rrv-.y 

^m«i 1 , ^ 2sm7r^(V 1/ V w/j ««oJ(7 \mj^ ' 



+m-i^ 



1+- 



(n + iyf (- a?y-i (1 - a?)** (ic. 



2 sin TT-er ^ J c 

Writing y = ti^ in this equality, we obtain 

IV 



1 liiL!j2l = _^ 

m 

where 2) denotes any simple contour in the plane of the complex variable y^ 
enclosing the point y = 0, and cutting the real axis in the point y«n. If 
now we make n increase without limit, we have 

r (z) = ^ — /"(- yY^'e-ydy, 
^ ^ 2 sin irzj ^ ^' ^ 

where the integral is taken along a curve commencing at positive infinity, 
circulating round the origin in the counter-clockwise direction, and returning 
to positive infinity again ; and in the integrand we must take (— yy^^ as 
equivalent to e}*^^ ^^ ^■^^ where the real value of log (— y) is to be taken when 
y is negative, and the logarithm is rendered one- valued by the stipulation 
that the variable is not to cross the real axis at any point on the positive side 
of the origin. 

Since 



r(5)r(l--^) = -r 



IT 



SmTT^ 

this result can be written in the form 



77)-2^/(-y>"'^"^^y- 



V{z) 

This theorem is valid for all values of z — in contrast to that found in the 
next article, which is true only for restricted values of the variable. 

Example 1. Bourgvs^s expressions for the Oamma-function, 
By a slight extension of the above proof, it is seen that 



101] THE GAMMA'FUNCTION. 183 

where the path of integration is restricted only to contain the origin and to be extended 
indefinitely at both ends in the direction of the negative part of the real axis ; the 
contour need not be closed. 

Take then as contoui* two lines inclined at an angle a to the axis of Xy passing through 
the origin, and a small circle round the origin. The integral round the small circle is zero 
when z has its real part comprised between and 1. The integration along the two 
lines gives the result 

^^'^^-^ rp*-V«-*sin(p8in«+^)c;p, 
which can be written in the form 

Tiz)^-. \-. rp'-^ef'^^'' Bin (p-hza) dp, 

^ ' sm zir (sm a)* J 



This formula is true for all values of a which are not less than -^ . Taking a equal to 
ir, we have the result 

r(«)=| p'-^e-Pdp, 

EaximpU 2. By taking for contour of integration a parabola with the origin as focus, 
shew that 

r(*)" 2,-ig-^^^ / «"**'(l+^*"*<5<>8[(2«-l)tan-ia?+^]cir. (Bourguet) 

101. Expression of T {z) as a definite integral, whose path of integration 
is real. 

We have, by the result of the preceding article. 

r (z) = =r^ { (5-V+ <«-« i<« <-y' dy. 

Take a path ABODE, commencing at the positive infinitely distant 
extremity of the real axis (which considered as initial point we denote by A\ 
proceeding close to the real axis until it arrives at the neighbourhood of the 
origin, describing a small circle BCD round the origin, and returning, close 
to the real axis, to positive inBuity again (which, considered as terminal 
point, we denote by E). With the conventions that have been made, the 
integral along the part AB of the path becomes 

i f^ 

I e^+(*-i)logy-t>(z-i)^y 

2 8m7r^;^ ^ 

in which log y is supposed to have its real determination. 

The part of the integral due to the small circle BCD is easily seen to be 
zero if the real part of z is positive. For the part of the integral due to DC, 
we have 



2 8in7rjrJo 



184 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 

Thus 

28m TT^r Jo 



or 



Jo 



This integral is called the Eulerian Integral of the Second Kind, It is 
frequently given as the definition of the Gamma-function : but for this 
purpose it is unsuited, since the integral exists only when the real part of g 
is positive. 

Example 1. Prove that when z is positive 

Example 2. Prove that 

Example 3. Prove that 

* e'^x^'^dx 



^X^'^dx^^\ 






(^+1)- • (^+2)* ' («+3)* • r(«);o «*-! ' 

102. Extension of the definite-integral expression to the case in which the 
argument of the Gamma-function is negative. 

The formula of the last article is no longer applicable when the argument 
z is negative. Saalschiitz has shewn however that, for negative arguments, 
an analogous theorem exists. This can be obtained in the following way. 

Consider the function 

r, (^)= JV> (^--1 +^- ^,+ ... + (- iy^'^)dx, 

where ^ is a negative number lying between the negative integers — k and 
-(A: + l). 

By partial integration we have, when ^ < — 1, 



r.w-[f («-■-.+.- ?,+...+(-!)"■ i;)] 



09^ 00 






102, 103] THE GAMMA-FUNCTION. 185 

The terms in the left-hand member which are not under the integral sign 
vanish, since (z + k) is negative and (-er + A + 1) is positive : so we have 

The same proof applies when z lies between and — 1, and leads to the 
result 

T(z+l)=^zr,{z) {0>z>-l). 

The last equation shews that, between the values and — 1 of ^, 

r,(z)-r(5). 

The preceding equation then shews that Fi (z) is the same as F (z) for all 
negative values of z less than — 1. Thus for all negative values of z, we have 
SaalschUtz's result 

F(^)=jV^(6-«-l+a:-|^j.h.:.+(-l)*+^|^)^^ 

where k is the integer next less than — z. 

Example. If a fimction P (ji)he such that for poeitiye values of fi we have 

and if for negative values of fi we define P^ (ja) by the equation 

where k is the integer next less than -^, shew that 

1 1 .,-.., 1 



AW=PW-- + fi-(;^)-... +(-!)*- ^7(;^ 



(Saalschiitz.) 



103. 0atL88* eaypression of the logarithmic derivate of the Gammorfunction 
as a definite integral. 

We shall next express the function -7- logF(r) as a definite integral, where 
z is supposed to be a positive real quantity. 

1 r* 

We have - = / er^dx, 

8 Jo 

Therefore loe « = I -de— 1 dx. 

^ Jis Jo on 



186 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 



Thus we have 



I e"*fi^"* log sds « I er's^^ds j 



— dx 



Jo ^ K Jo Jo ) 



r<.).rw/;^|<r--,ji^}. 

This equation is due to Dirichlet. 

Writing 1 -♦- a? = e* in the second term of the integral, and a? = ^ in the 
first term, we have 

which is Qauss' expression of -r- log r(^) as a definite integral. 



^l0gr(«)- / I— Irr-^J rf^ . (Gauss.) 



Example 1. Prove that 

dz 

Example 2. Prove that 

d. ^,, , 1 P . r«(l-a) a(l-a)(2-a)^ "1 



^■-/•■{q--} 



104. Binet*8 expression of log T (z) in terms of a definite integral. 

Binet* has given an expression for log r(^), which is of great importance 
as shewing the way in which log F (z) increases as z becomes very large ; his 
result will be used later in the derivation of the asymptotic expansion 
of r {z). 

We have by the last article (z being supposed real and positive) 






r 



changing -^ to ^ + 1, we have 



5"«I'(' + ')-/"(t-*^)*- 



Now f er^dt^^, 

Jo ^ 



* JawmaX de Vie. Polyt xvi. (1S39), pp. 128—343. 



104] 



and 



THE OAMUA-FUNCnON. 



187 



/, 



— (e « 



t 



Jo Jo 

Jo y 

= log z. 



Therefore 



|iogr(.-Hi)=^+iog.+/;d*{-?^^-?:^ 



-< — /?-t« 



,-e 



-te ^ 



e 



■^ t 6«-l 



or 



Integrate for z between the limits 1 and z ; so 

iogr(z+i).|iog^+^(iog^-i)+j"J{^-l+l 
i<«r(.)-(.-l)iog.-.+/;{j^,-l+^}r? 



t 



dt 



dt 



*'-fA^A*-^r'' ('> 



The first of these integrals can be otherwise expressed in the following 



way. 



We have* 



-l"^2 t Jo e^'^-1 ' 



Multipljong both sides of this equation by e~^ dt and integrating with 
respect to t from zero to infinity, we have 



^Jo (u' + 



udu 



(w*+i>*)(e^»-l)* 
Integrating this equation from j> » ^ to /> = oo , we have 



*• 



i:?*(i^i4-i)- 



tan" 



■ © 



dt 



6^-1 



(2). 



* A proof of this equation can be foand by making k infinite in the equation 

2 -rs-i — 0=2 S I «-*•»« sin (ttt) dt*. 



188 



TRANSCENDENTAL FUNCnONS. 



Thus equation (1) becomes 



logr{z)^(^z~^logZ'Z'^2 



tan"" 



■©"• 



e«'»-l 



^^ Jo V-l t^2 



--dt 
t 



Now write z^^^ equation (1): since 



r© 



"», 



we obtain 



1 , 1 r* ( 1 11 



6 



-i< 



t 



dt 



-rA^.-w^T^ 



t 



Write ^ for t in the last integral. Thus 



or 



= 



= 



g log IT 
5 log IT 



_i_r{_i 1 r 

2 Jo V-1 ^^^I'^t 

1 f» f -1 e-f" 

2 Jo te'-l''" « , 






dt, 



e-^) dt 
t 



Adding this to equation (3), we obtain 



' 1\ 
logr(«) = lir-2Jlog«-« + 2 



[chap. IX. 



tan~M-)du , , 



.(3). 



+Jo « v-i t (^-i^ t 2} ^*^- 



The last integral is 



rdt(l , tr* 



-e-»«l 



t 



or 



/ft J ^ 



JO 



t 



or 



I log X dx, 



* This ftrtifioe it dae to Pringsheim, MatK Ann, xxxi. 



105] THE GAMMA-FUNCTION. 18^ 

or . 'l^ogl-^l. 

Substituting this in equation (4), we obtain 



logr(^) = ^^-2Jlog^-2: + 2log(27r) 








g2iru_l 



This is Binet's formula for log r(z)] ss z increases indefinitely, the last 
integral diminishes indefinitely, and so the remaining terms furnish an 
approximate expression for logr(^:) when z is large. 

Example, Prove that 

log r(«)=(«-i) log«-«+i log (2ir)+ J(«), 
where J (z) is given by the absolutely convergent series 

•^ ('^"* |m^ l"*" 2 (2r+l)V+2)'*' 3 (^+l)(« + 2)(«+3) ■*"•••/' 

in which 

^i~J> ^2~J> ^"fo> ^4"*Tcr> 
and generally 

c^z^j (x+l)(a?+2)...(a?+n-l)(2x-l)a7ctr. (Binet) 

106. The Eulerian Integral of the First Kind, 

The name Eulerian Integral of the First Kiixd was given by Binet to the- 
integral 

jB(p, ?)= [ x^^ {I - x)^-^ dx, 
Jo 

which was first studied by Euler and Legendre. In this integral, the real 
parts of p and q are supposed to be positive ; and x^~^, (1 — x)^^ are to be 
understood to mean those values of e^^^^**^* and e<^*^**^^""*^ which correspond 
to the real determinations of the logarithms. 

With these stipulations, it is easily seen that the integral exists, since the 
infinity of the integrand is of less than the first order at the two extremities, 
of the path of integration. 

We have, on writing (1 - a?) for x, 

B{p.q)=B{q,p). 

Also 



or 



It 



190 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 

Also 

Jo Jo 

= Bip+l,q) + B{p,q+l). 
Combining these results we obtain the formula 

Example 1. Prove that if n is a positive integer, 

r>/ i-iv * 1.2... tl 

^(^'''+l)-p(^+l)...(p+n)- 
Example 2. Prove that 

106. Expression of the Evlerian Integral of the first kind in terms of 
the Oamma-function, 

We shall now establish the important theorem 

^. ^ r(m)r(n) 

B (m, n) = 't^^ — -/ . 

^ * ^ r (m + n) 

To prove this, we have 

/•OO /•OO 

r (m) r (n) = I e-%^*-^da; x | e-yy«-^ dy 

JO ./o 

(writing a^ for a:, and y" for y) 

Jo Jo 



4 / e-^'^-^^^af^^y^-^dxdy 

J J 



(writing r cos ^ for x, and r sin for y) 






= 4 I f e-*«r*("*+n)-i cos*^^ sin«^-^ ^ dr dtf 

^0 Jo 

= r (m + n) 2 I cos«^» ^ sin*^i ^ d^ 

(putting cos* 0^u) = r (m + n) -B (m, n). 

This result connects the Eulerian Integral of the first kind with the 
Gamma-function. 



106 — 108] THE QAMMA-FUNCnON. 191 

Example. Prove that 

(Cambridge Mathematical Tripos, Part I., 1894.) 

107. Evahiation of trigonometrical integrals in terms of the Oamntar- 
function. 

We can now evaluate the integral 

w 

•a 



I cos*'*~'a?8in**"^a:da?, 

Jo 



where m and n are not restricted to be integral, but have their real parts 
positive. 

For writing sin* a?= ^, we have 

r* 1 P ^-1 5.1 



1 r/^ n\ 

- 2 , (^.) 



The well-known elementary formulae for the case in which m and n are 
integers can be at once derived from this 

Example. Prove that when | iE: | < 1, 

pcoS^8in*^flW V 2 y \ 2 y P cos*»-"*^(f ^ 
jo O^siE^^jT" ^- /m+n+l\ -/<> (^.^^i^,^)!^- 

(Trinity College Examination, 1898.) 

108. DirichleSs multiple integrals. 
We shall now shew how the integral 

may be reduced to a simple integral, where / is an arbitrary function of 
its argument, and the integration is extended over all the systems of 
positive values of the variables x, y, z, which satisfy the inequality 



192 



TRANSCENDENTAL FUNCTIONS. 



[chap. Et. 



Write 



111 
a?«(L»i*, y^hyf, z^cz{'. 



Then the integral takes the form 



/ = 



a^b^C 



a/87 



- jjJA^ + yi + ^) x^^^-^yi^^-^z^^-^dx^dy^dzu 



where the integration is now taken over all the systems of positive values of 
the variables ah, yi, ^1, which satisfy the inequality 



^ + yi+'«i^i. 



Now let 



/i = ^+yi + ^i-f=0, 

be three equations defining new variables f, 1;, f. 



-1 



a(/i,/2,/s) 



Then 



9(^1, yi, ^1) 



-^ 
-1;? 












1 





1 
1 




1 
1 
1 



=fV 



The field of integration is clearly such that the new variables f , 1;, f, each 
vary from to 1. 

Thus 

/ = ^^^^^ C f ^ f V(f ) P'+^»+''»-^ (1 - ^)^'-»i7«i-^'-^ (1 - O^^-'^-'d^dvd^ 
otp7 JoJoJo 






a^7 



-50),, 9. + r,)5(g.. r,)f /(f)P-^''+"-'df 

Jo 



"a^7 



\a i8 7> 



The multiple integral is reduced to a simple integral. 

It is easily seen that this method of evaluation can be applied to multiple 
integrals of a similar form in any number of variables. 



109] THE GAMMA-FUNCTION. 193 

Example 1. Shew that the moment of inertia of a homogeneous ellipsoid of unit density, 
taken about the axis of «, is 

-i(a«+i^«)fra5c, 

where a, 6, c are the semi-axes. 

Example 2. Shew that the area of the epicycloid ^ -)-y^s= ^ is |ir^. 

Note, Dirichlet's integrals can also be evaluated by performing on the variables the 
substitution 

a?i«sf^ sin* $1 sin* ^„ 

yi «= r* sin* ^j cos* ^j, 

«i»!r*cos*^i, 

leading to the same result as above ; in the case of an integral with n variables the 
corresponding substitution would be 

a?i «r* sin* $1 sin* 6^ ••• sin* ^»- j, etc. 

109. The asymptotic expansion of the logarithm of the Oamma-function 
{Stirling's series). 

We now proceed to 6nd an expansion which asymptotically (§ 88) 
represents the function logr(i^), and is actually used in the calculation of 
the Qamma-function. 

For simplicity, we shall consider only real positive values of the argument 
z. For a proof and discussion of the expansion when z has complex values 
the student is referred to a memoir by Stieltjes*. 

From Binet's expression for log F (z) (§ 104), we have 

logr(^) = (^-5)log^-^ + |log(27r) + <^(^), 



tan"' - ax 
where ^(^) = 2/ ^^^ 



r) = 2J 

* A 



Now tan-- = ----+---... 



(-1 y»-> ^-^ (-1) ^ r* t^dt 

'*"(2n-l)~^^-i"^ z^^^ Jo t^ + z* 



Substituting this in the integral, and remembering that 

a?**"^ dx Br. 



/ 



e»»*-l 4n' 



* Lumvt2Ze*< J<mmal (4), v. pp. 425—444 (1889). 
W. A. 13 



194 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 

where jBj, fij, ... are the BernouUian numbers, we have 

Let us now find approximately the magnitude of the last term when z is 

very large. 

r* dx (^ t^dt 
The quantity j^__J^__^ 

is less than 



'''' ^1n+l)^«Jo 






or 



4(n4-l)(2n+l)z«* 

If now any value of n be taken, it is clear that this quantity can be 
made as small as we please by taking z sufficiently large. 

It ollows that the quantity 



|<PV^; r^i 2r(2r-l)^»-iJ 



can be made as small as we please by taking a sufficiently large value 
for z ; and therefore (§ 88) the series 

-r 



l,2.z 3.4.^ ■ 5,Q.z' 

is the asymptotic expansion of the function <f> (z) for large real positive 
values of z. 

We see therefore that the series 

l\i If /« N 5 (-ly-^Br 1 



ar— 1 



is the asymptotic expansion of the function logr(z) for large real positive 
values of z. This is generally known as Stirling's series, 

110. Asymptotic expansion of the Oamma-function, 

Forming the exponentials of both members of the equation just found, 
we have 

S B S 

r (z) = e-^z'-i(2v)i^''~^^^^^'"" , 



110] THE GAMMA-FUNCTION. 195 

or 

r (^) = «--^-i (2,r)* |l + ^ + § + . . .1 , 

where Cj = z-— ^ , C, = -^ , etc. 

Substituting the numerical values of the Bemoullian numbers, the 
formula becomes 

n,z)-e z- \''^}'y-+i2z^2{12zy 30(12^)' 120(12^)* "'j' 

This is the asymptotic expansion of the Oammorfunction, In conjunction 
with the formula r(l -{-z)^ zT{z), it is very useful for the purpose of com- 
puting the numerical value of the function. 

Tables of the function log r {z\ correct to 12 decimal places, for values of t between 
1 and 2, were constructed in this way by Legendre, and published in his Exercices de 
Calcul Integral, Tome il p. 85, in 1817. 



Miscellaneous Examples. 



1. Shew that 



<-.>(-l)('-3)(-l)--7(:;^- 

(Trinity College Examination, 1897.) 
2. If o-n be the sum of the n first terms of a divergent series 



- + - + - + ..., 
«i Oa «8 

shew that the series 

1 1 1 



Ojo-i OfO-f a^^ 
is divergent. 

If the squares of the terms of the latter series form a convergent series, shew that a 
function O (l+z) can be defined by the equation 



6^(1+^)= Limit 



<rn' 



and shew that 






tf(T+7)-^°.{(i+i-J «■'••'-}> 



where e is a constant. (Ceearo.) 

13—2 



196 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 

3. Prove that 



dz Jo !-«-« 



-f"{(i+«)-'-a+«)-}T-r. 

Jo fl 

where y is Eider's constant. 

4 Prove that 



5. Prove that 



r («) = Limit n'B («, n). 






6. Prove that, when ^ > 1, 

^(p, j)+^(p+l, j)+^(p+2, ?)+...-5(p, 9-1). 

7. Prove that 

Biscay q) gjr a(a + l)9(g + l) 

^(i>i?) '*"l>+?"^1.2.(p + j)(p+j + l)"^-- 

8. Prove that 

^tP, q)B(p+q, r)^B(q, r) B(q + r, p). (Euler.) 

9. Prove that 

log r (*)=(!-«) log ir+y(i-«)-i log sin «irH — 2 -^ — sin2n«9r. 

IT n-i n 

(Kummer.) 

10. Prove that 

I co8'>+«"*wco8(p-g)w(iM—; — ; Tv-r^rx:; ,-„-, r. (Cauchv.) 



11. Prove that 



..»i.(«,)-w(^)*/;'-!;=!^az|L'^.. m^-> 



12. Prove that 



2»(p.i'+«) 2J— V+2(2^1) + 2".M2p+iK2^3) + "r ^ ^^ 



13. Prove that 

r 



r 

14. Prove that 



(j»+i)"lP 4p0>+l)^2.4».^0j+l)(p+2)^-; • 



\ V(p) ] 2 t 2(2p+l) 2.4.(2p+3)(2p+6)^-r 



MISC. EXS.] THE GAMMA-FUNCTION. 197 

16. Prove that 

(Binet.) 

16. Prove that 

where y is Eider's constant. (Legendre.) 

17. Prove that 

B(p,p)B{p+i,p+i)^,^^. (Binet) 



18. If 



shew that 



/ \ogT{z)dz^u, 



du , 

and hence (or otherwise) that 

ti^orlog J?— ^+^log29r. (Raabe.) 



19. Prove that, for all values of t except negative real values, 

logr«-(f-i)log^-«+ilog(2^)+ i rTTl- 



Sin %nirx 

(Bourguet.) 



n^r 



20. Prove that 



,_ r(a-hl)r(a+6+c4-l) Pfia-f^KLif!)^^ 
*'^r(a+6+l)r(a+c+l)"jo~;^ ' l ^- 



(l-^)log- 



21. Prove that 



ri^«-i-^-i ^ V 2 / \2/ 

Jo(l-H^)loga?''^^ /a\ //3-Hy 



(Kummer.) 



22. Prove that 



, r(a+6+l)r(a-Hc+l)r(6H-c-H) f Ml -^)a-^)(l-^) ^_ 

'^r(a+l)r(6+l)r(c+l)r(a+6i-c+l)";o ,, M 1 



(l-x)log- 



23. When a? is positive, shew that 



- T{x) • 2nl 1 



v;;-^w 2 



(Cambridge Mathematical Tripos, Part I, 1897.) 

24. If a is positive, shew that 

r(g)r(a+l) ^ « (~ l)''a (g- 1) (a- 2) ... {a-n) 1 
r(«+a) ^ »=o n\ « + »* 



198 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 

25. Shew that 

26. The curve f^ «= 2** ~^ cosing is composed of m equal closed loops. Shew that the 
length of the arc of half of one of the loops is 

1 f' -i-1 

2^"^.—. / (cosarV dx, 
and hence that the total perimeter of the curve is 

27. Prove that 

logr(«)=(«-i)log«-«+ilog(2fl-) 



28. Prove that 



29. Prove that 






30. Prove that 

a- a* 



log r («+a)=log r («)+a log « - 



2« 



a / a(l — a)<ia- / a{\^a)da 
Jo Jo 

2s (2+1) 

a/ a(l-a)(2-a)flfot- / a(l-a)(2-a)tia 

3«(«+l)(«+2) 



31. Prove that 



32. Prove that 



5(p.?)-(^r,!.i(2-)*»*(p.«, 



MISC. EXS.] 
where 



THE OAMMA-FUNCTION. 



199 



M(p,q)^2pj*^ 



dt 



e^i^-l 






(p+?)r 



and 



p*— ^+^+/>S'. 



33. Expand 



{r(a)}-i 



as a series of ascending powers of a. 



(Various evaluations of the coefficients in this expansion have heen given by Bourguet, 
Bull, des Sci. Math, v. (1881), p. 43; Boiu^uet, Acta Math, ii. (1883), p. 261 ; Schlttmilch, 
Zeit$ohrift fiir Math, xxv. (1880), pp. 35, 851.) 



34. Shew that 



and 



where 



r^^-a«oos 6a?<iir-co8 {(m + 1) («■ - <^)} ?^-^^^^ 

r^^—xsin &p<iir-sin {(m+ 1) (ir - 4)) ^^j- , 
- a-k-bi^r (cos 0+t sin ff>). 



35. If 



I e-'tf^idty 



shew that 



and 



Pi^)--] 



^^^^ X l!a?+l'*"2la?+2 3!^+3"*"*"' 
P(x+l)-arP(ar)-i. 



36. Prove that 



d i-, r(z+ar) X ,x {x-l) . x{x-'\){x-'%) 

5^*"* r(z) "« ^«(«+i)"*"^ «(«+i)(«+2) "^•••* 



37. If a is negative, and if 
where v is an integer and a is positive, shew that 



r^r) r (a) 
r( 






where 



(-l)»(a-l)(a-2)...(a-n)^ 



0.(,).gW-0(-«), 



(Hermite.) 



200 TRANSCENDENTAL FUNCTION& [CHAP. IX. 

38. When - oo <a< 1, shew that 

r{x)T(a-x) % K ; R* 

r(a) ",!, ^+;i-,*,^::^:rn' 

^^ (-l)»*a(aH-l)... (aH-n-1) 

39. When a > 1, and y and a are reepectively the integral and fractional parts of 
a, shew that 

T{x)T{a-x) ^ J G{x)p^ • Q{x)p^^^ 

L^— a JP— a— I 4?— a — v+lj 

where 

*«-('-!)('-^)-('-,-T^) 

and 

_( - l)**o(a+l) ...(a+n-l) 

40. If Pi, p^y ... py are the roots of the equation 

p»' + aip''-i+...+aK«0, 



shew that 



• f/ X X* ' ^^ \ *!*) 



,r(l)a.Xj^^^^ 



T (z- pix)r (z- p^) ...r {z- p^)' 

41. If a and b are real and positive, prove that 

/ e"i'V-iw«-*-irfttrfv»r(a)r(6). 

42. By taking as contour of integration a parabola with its vertex at the origin, derive 
from the formula 

^W = s^ l^^^^dz 

^ ' 2tsma9ry 

the result 

T{a)'^^^^f e-'^x^'i (I +j^)-^ [3 sin {x+acot'^i-x)} 

+8in {x+(a - 2) cot-» {'-x)}]dx. 

(Bourguet.) 

43. Prove that, when 1 < « < 2, 

/*sin6^ - __ ^"* «• 
and when < « < 1, 



sm- 



pcosftg? . _ f-^ 

Jo A- 2r(«; 



«■ 



) tr«' 



MISC. EXS.] THE OAMMA-FUNCTION. 201 



44. Shew that 

w 






46. If 

2« 



t7» 



('-!) 



and 

s 

2« 



F-r 



and if a function F(x) be defined "by the equation 

shew (1) that F(x) satisfies the equation 

(2) that for all positive integral values of x, 

F(x)~T{x\ 

(3) that F{x) is regular for all finite values of x, 

(4) that 

46. Prove that the function O (x), defined by the equation 

6f(a?+l)=(2ir)«e « « n fl+lj e» , 
has the properties expressed by the equations 

log ^j^- , — (a» I «-^cotir:J?c2r-a?log2tr, 

[ (g-l)(a;-2 ) «-l r Q +itn 

(n+i)-i- (r(«+i)}.- in iij±;ij. 

(Alexerewsky.) 

47. If « is a positive quantity (not necessarily integral), and « is a real quantity 
between - ^ and ^ , shew that 



COS* 



1 r(«+i) f. « „ . «(#-2) . . \ 



and draw graphs of the series and of the function cos's. 



202 TRANSCENDENTAL FUNCTTIONS. [CHAP. IX. 

48. Obtain the expansion 



and find the values of x for which it is applicable. 
49. Prove that 



(Cauchy.) 



i.(-5)' '^^" 



60. If 






('•*^-rV)/o 



where | ^ | < I and the real part of or is positive, shew that 

and 

Limit (1 - x)^-* f («, ^) = r (1 - *). 

ae-l 

61. If Xf Wy and « be real, and < t9 < 1, and « > 1, and if 



shew that 



and 



00 ^iri* 



*(«.,;r,l-,).^J ^'' I. (Lerch.) 



62. If 



shew that 






tf'-l ' 



(3) r(|) n-iCW-rQ^^ ,r*Tf(l-,). 

63. Let the function <f>i''>(x) be defined by the equation 



(-!)•*«( 



I n(i-«— ^) 



where 5 is an integer ^ m, the function x (0 ^^ defined by the equation 
and the quantities a^ are constants whose real part is positive. 



MISC. EX&] THE QABfMA-FUNCTION. 203 

Shew that ^t*) (x) can be expressed by the series 

<^«(^)-2/W(jp+M^); 

where t9«>ZXrart 

and where 

(-i)'/«(«)-j'"x(0 «•«-"*• 

Shew also that ^t*) (x) satisfies the functional equation 



M.»' 



Shew further that when ;^(r)«Bl, <f>i')(x) becomes a function ^^(x\ which has the 
multiplication-theorem 

where all the quantities X vary from to (n- 1). 

(Pincherle.) 

64. If 



where 






/n\ n! 

W"rl (n-r)!' 



shew that 

. riy)r(sf'-x+n)T(x+v)T{v+n) 

/nwy, «^; r(y-i?)r(y+n)r(t^)r(a?+v+n)' 

and that 

r(y)r(a?+v) 



(y-x-l)r(-n)r(a?+l)r(y+t;+n-l)' 

(Saalschiits.) 



CHAPTER X. 
Lbgendbb Functions. 

111. Definition of Legendre polynomials. 
The expression (1 — 2zh + /i*)"^ 

can, when | A { is sufficiently small, be expanded by the multinomial theorem 
as a series of ascending powers of h, in the form 

where Pi {z) = Zy 

-P« (^) = ^^-^— ^ . etc. 

The expressions Pi{z), P,(^) ..., which are clearly all polynomials in z, 
are known as Legendre polynomials, Pn (z) is called the Legendre polynomial 
of order n. 

It will appear later (§ 116) that these polynomials are particular oases of a more 
extended class of functions, known as Legendre functions. 

Example 1. Prove that 

P„(coe^)=^ — r^coseC*-^^^ ,/ ^' 
" ^ n ! a (cot B)^ 

(Cambridge Mathematical Tripos, Part II, 1893.) 

Let ^ be an angle such that 



Then 



(1- 


-2A 


cos^+A«)" 


4_8in ff 
"~ sin ^ * 










cot^« -r-=-^- 


* 










coe^-^ 
~ sin^ 












■scot^-A 


cosec 


B. 



Ill, 112] LBGENDBE FUNCTIONa 205 

Therefore by Taylor's theorem we have 

• iK « ( - A cosec ^)» €?* (sin ^) 

sin ff ^* ;; « V . ^x" • 

n n\ (f(oot^)»' 

or 

/, or ii.nv-* ^(-A)*coseo»+»^flP«(8m^) 
(l-2Acoe^+A») *=S^-^ir! d^r 

Equating coefficients of A*, we obtain the required result. 

Example 2. Shew that 



For 



/>-©'■ 



Therefore 



Thus 



(l-2A*+A«)-*=ir-* r «-(i-«'+*^««cft=ir-* f * tf-(i-'«)««6-(»-*y««(&. 

y —00 J —00 



whence the result follows. 

Example 3. By equating coefficients of powers of A in the expansion 

(l-2Aco8^+A*)* \ * 2.4 / 

X (n-iAe-«+^A«d-««+...), 



shew that 



112. SchldfiVs integral for Pn{z\ 

Let h be any quantity which is not greater than the radius of convergence 



00 



of the series 2 h^Pn {z\ 



Then (1 — 2zh + A*)~* can be expanded as the series 

1+AP^(^) + A'P«(^)+A»P3(^)+.... 

But (1 - 2zh + A')~* is the residue, at the pole 

1 (1 - 2zh + /i')* 
^"h" h 

of the function — 2A""* "^ I ^ "" t J Ti r • 



206 TEANSCENDENTAL FUNCTIONS. [CHAP. X. 

Now the last expression has two poles, namely at the points 

1 (1 - 2zh + h*^ 

and t^Ud^^^^, 

ft n 

When h is very small, the former of these poles is close to the point 
t = z, while the second pole is in the infinitely distant part of the plane. 
Therefore, if (7 be a contour in the ^-plane, including the point z, the former 
pole only is contained within C when h is not large, and so we have 






di. 



Equating coefficients of A", we have the result 

which is called Schldflis integral-formula for the Legendre polynomials 

113. Rodrigues* formula for the Legendre polynomials. 
From Schlafli's integral 






dt 



we immediately deduce, by the theorem of § 38, the result 

which is called Rodrigvss formula, 

114. Legendre'a differential equxttion. 

We shall now prove that the function y = Pn (-^) is a solution of the 
differential equation 

which is called Legendre's differential equation of order n. 

* Schlafli, Ueber die beiden Heine*$cheii Kugelfunctionen ; Bern, ISSl. 



113—116] 



LEOENDRE FUNCTIONS. 



207 



For on substituting Schlafli's integral, we have 



ds^ 



dz 



27n 



and this integral is zero, since the function (<* — 1)**"*"^ {t — ^)~"~' resumes its 
original value after describing the contour (7. The Legendre pol3n[iomial 
therefore satisfies the differential equation. 

The differential equation can clearly be written in the alternative form 

116. The integral-properties of the Legendre polynomials. 
We shall now shew that 



and that 



j[Pm{z)Pn{z)dz = 0, 



if m and n are positive integers and m is not equal to n. 

dPn] 



For since 



we have 






+ (m-n)(m + n-l)\ PnPndz^O. 

■ —1 

Integrating by parts, this equation gives 
- (m - n) (m + n - !)/'-?». («) -P» («) dz 

-['a-^){p„(.)^-P„w^)}; 

which shews that the integral 

j' P„(z)Pn{z)dz 

has the value zero when m is not equal to n. 



= 0, 



208 TRANSCENDENTAL FUNCTIONS. [CHAP. X. 

To establish the second part of the theorem, let the equation 

be squared, and the resulting equation integrated between the limits — 1 and 
+ 1 ; using the result already proved in the first part of the theorem, we thus 
obtain 



or 






Equating coeflScients of h^ in this equality, we have 

which is the desired result. 

Example 1. Prove that, if m is not equal to n, 

x{l +(-!)*+«}. 
(Cambridge Mathematical Tripoe, Part I, 1897.) 

Example 2. Prove that 

j-i dzr d^ ^^ '^ '^^'''^ (27i + l)(n-r)!' 

according as m, n are unequal or equal. 

(Cambridge Mathematical Tripos, Part I, 1893.) 

116. Legendre functions. 

Hitherto we have supposed that the index n of Pn{z) is a positive 
integer; in fact, Pn{z) has not been defined except when n is a positive 
integer. We shall now see how the definition can be extended so as to 
furnish a definition of Pn{^)» even when n is not integral. 

An analogy can be drawn from the theory of the Gamma-function. The expression 
zl as ordinarily defined (viz. as « («— 1) (*-2)...2. 1) has a meaning only for positive 
integral values of z; but when the Gamma-function has been introduced, z\ can be defined 
to be r (2 + 1)) and so a function z\ will exist for all values of z. 

Referring to § 114, we see the differential equation 



116] LEQENDRS FUNCTIONS. 209 

is satisfied by the expression • 

even when n is not a positive integer, provided that (7 is a contour such that 

the function 

(^ - l)n-fi 

{t - xr)»+a 

resumes its original value after describing C. 

Suppose then that n is no longer taken to be a positive integer. 

Now the function (f*— 1)*+* (t — xr)-**-« has three singularities, namely the 
points t^l,t^^l,t»z; and it is clear that after describing a small closed 
contour enclosing the point ^^l, the function resumes its original value 
multiplied by e»»»<»+i) ; while after describing a small closed contour enclosing 
the point t = z, the function resumes its original value multiplied by 

If therefore (7 be a simple contour enclosing the points t^ I and t ^ z, 
but not enclosing the point ^=: — 1, then the function 

(e» - l)n+i 



Ji+a 



(t-z) 

will after describing C resume its original value multiplied by e"***, i.e. it will 
resume its original value. Hence whatever n he, Hie Legendrian differential 
equation of order w, 

is satisfied hy the expression 

where is a simple contour in the t-plane enclosing the points t^l and t^g, 
hut not enclosing the point ^ » — 1. 

This expression will he denoted hy P» {z\ and will he termed the Legendre 
function of the first kind and of order n. 

We have thus obtained a definition of Pn {z) which is valid even when n 
is not integral. 

The Legendre function is a mere polynomial when n is integral, but is 
a new transcendental function when w is not integral ; just as F {z) is the 
polynomial (^-* 1)1 when z is integral, but is a transcendental function when 
z is not integral. 

W. A. 1* 






210 TRANSCENDENTAL FUNCTIONS. [CHAP. X. 

We shall suppose the many-valued function ^, which occurs in the 
defining integral, to have the value 1 when z is equal to 1, and when z 
is not equal to 1 to have that value which would be obtained by con- 
tinuation along a rectilinear path from the point 1 to the point z, 

117. The Recurrence-formulae, 

We proceed to establish a group of formulae which connect Legendre 
functions of different orders. 

We have by § 116, for all real or complex values of w, 

m 

Integrating by parts, we have 

27ri;e2»-(<-«)" 
and hence we have 

f „(.)-.P^. (^)- 2^/, 2^^(7-1;^. ^^ <^)- 

Differentiating this equality, we obtain 

—dz '~~d'z ■^"-' ^'^ -~2^lc S^'it-zr*^ - ^" ^^^»-' ^^^' 

80 

^>-.%i^>-nP„_.(.) (I). 

This is the first of the required formulae. 
Next, from the identity 

we deduce 



or 



r (f'-l)»-> ^,, f 2 Kf - 1) + 1} (n - 1) (f - If-' dt 

[ (n - 1) {(t -z) + z] (f - ly-' ^, 



117] LEGENOEE FUNCTIONS. 211 

or 



or 






(n-i)^r o*-iy'-^ 






27n; ./c2'*-H^-'8^V 
or, by formula (A) above and Schlafli's formula, 

= n{PnW--^Pn-iW}+(w-l)Pn-.(^)-(^-l)^Pn-iW, 

or 

nPnW-(2n-l)5Pn-i(^) + (n-l)Pn^(^) = (H), 

a relation connecting three Legendre functions of consecutive orders. This 
is the second of the required formulae. 

Other formulae can be deduced from (I) and (II) in the following way : 

Differentiating (II), we have 

"" --^— -(2n^l)^ d^ "^^^"^^"d^ =(2n-l)P^,(4 

Substituting for 

dPn (Z) 

dz 
from (I), we have 

f dPn-x^) p ,1 dPn-x(^)^,^ ,, dPn->(^) 

= (2n-l)P„_,(^), 
or 

-(n-l)r^^g<l> + (»-l)^^g^^ = -(n-l)'P^,(4 
or 

^ —Tz -^^ =(n-l)P„-,(4 

Changing (n — 1) to n in this equality, we have 

Next, changing n to (n + 1) in (I), we have 

14—2 



r^^^BiB^BR 



212 TRANSCBNDENTAL FUNCTIONS. [CHAP. X. 

Adding this to (III), we have 

^-P^^(^) - ^-P^^(^) ,(2n + l)Pn(^) (IV). 

Lastly, combining (I) and (III), we obtain the result 

{ii>-l)^^^nzPniz)-nP^r{z) (V). 

The formulae (I) to (V) are called the recurrence'/ormtdde. 

The above proof holds whether n is an integer or not, Le. it is applicable to the general 
Legendre functions. Another proof which, however, only applies to the case when n is a 
positive integer (i.e. is only applicable to the L^;endre polynomials) is as follows : 

Write 

Then equating coefficients of powers of A in the equality 

(l-2Az+A«)|^=(«-A)r, 

we have 

nP,W-(2n-l)«P^_iW+(n-l)P,,.,W-0, 

which is the formula (II). 

Similarly, equating coefficients of powers of A in the equality 



we have 









which is the formula (III). The others can be deduced from these. 
Eixtmple, Shew that, for all values of n, 

(2n+3)P«,^i-(2n+l)P.«=^{z(P,«+P«,+i)-2P»A+J. 

(Hargreaves.) 
For 

^{^W+i^n*i)-2P,P,^J 
=P.«+P«..,+2.P.5^+2.P..,%^-2P.%.2P.,,f^ 

=P,«+i»,+,+2P,(-n-l)i>.+2P,^i(n+l)i'.+ , 
(as is seen by using formulae (I) and (III)) 

-(2»+3)i«,^,-(2n+l)P,«, 
which is the required result 



118] LEGENDRE FUNCTIONS. 218 

118. EvaiuaUon of the integral-expression for Pn (z), as a power-series. 

When n is a positive integer, we have seen that Pn {z) is a polynomial in 
z. When n is not a positive integer, however, P^ {z) is not a polynomial ; 
and as P^ (z) is not a regular function of z for all finite values of z unless n 
is integral, it follows that no power-series exists which represents P^ (z) for 
all finite values of z, when n is not integral. In order to find a power-series 
capable of representing Pn(z), we must therefore make some supposition 
regarding the part of the 5-plane on which the point z lies. We shall suppose 
that z lies within a circle of radius 2, whose centre is the point 1 ; so that 

|1-^|<2. 

As the contour C of § 116 was subject ooly to the condition of enclosiug 
^ a> 1 and t^z without enclosing t » - 1, it is clear that we can choose it so 
as to lie entirely within the circle of centre 1 and radius 2 in the f-plane, 
ie. to be such that the inequality 1 1 * f | < 2 is satisfied for all points t on C. 

Now write ^— 1 = (^ — l)t^. When t describes the contour C, the point 
representing the variable u will describe a contour 7 on the t£-plane ; since 

C encloses the points t^z and ^ » 1, 7 will enclose the points u^l and u^O; 

2 
and since |1 — <| < 2, we shall have \u\< . ^ . for all points u on 7. 

Then changing the variable of integration from t to u ih the integral 
which represents Pn (z), we have 



^(^•^V-T^'*^--^)-^^ 



2wi.y 

2 

Since \u\ < -, 77 we can expand this in the form 



du. 



Now on integrating by parts, we have the result 

J y Ly W J Tt J y 

The first expression on the right-hand side is zero, and so we have 
I !*•'+*• (u-l)-*^»dw«^^^[ w«'+»-»(w-l)-~-*(u-l)dw 

n Jy ^ ' n Jy 

or 

I w'+« (u - !)-*-> du = ^^i^ I w'"+«-^ {u - 1)-«-* du. 



214 TRANSCENDENTAL FUNCTIONS. [CHAP. X. 

Therefore 

Now transform the integral on the right-hand side, by writing u ss 



v-l* 

r f t?*dt; 

The integral I w*(tt — l)~*~*(iu becomes — I — -r , where the integration 

has now to be taken in the positive sense round a contour 8 enclosing the 
points t; » and t; «= oo , but not enclosing the point v^l. This integral can 

be replaced by + i --— j , where the integration has to be taken in the 

positive sense round a contour S^ enclosing the point t;= 1, but not enclosing 
the points v^O, or t; ae oo (since the integrand has no singularities in the 
region between the contours B and S^). The contour B' can now be diminished 
until it becomes an infinitesimal circle surrounding the point t; = 1. The 

value of the integral is then 1* I -— ^ , where the integration is taken round 

this contour; or 2m, since the many- valued function v* has been taken 
to have the meaning 1 at the point t; = 1. We thus have 






r + n r — 1 + n 1 + n 



2mjy ^ -/ -- ^ • ^.1 ••• 1 ' 

and on substituting this in the expression already found for Pn(^)i we 
obtain 

p , X 1 _^ S n(n-l)...(n-r-H) (r-hn)(r-l+n)...(l-f n) fz-lV 

an expansion of Pn (z) as a series of powers of ('2^—1). 
If now, as in § 14, a series of the form 

^^l.c^^ 1.2c(c-hl) ^+- 

(a hypergeometric series) be denoted by 

F(a, 6, c, z\ 
then the expansion can be written in the form 

Pn(r)-J'(-n, n + 1, 1, ^^). 

This is the required expression for P^ (z) as an infinite series. It is valid 
at all points z within the circle whose equation is |1 — ir| < 2. 



119] LEGENBBE FUNCnON& 215 

Corollary, Since this series is clearly unaffected when n is changed to 
— n — 1, we have 

Note. When n is a positive integer, the above series terminates and gives the expression 

1 — « 
of P» («) as a polynomial in —^ . 

119. Laplace's integral-expressuyn for P^ {z). 

We shall next shew that, for all values of n and for certain v^alues of z, 
the Legendre function Pn{z) can be represented by the integral (called 
Laplace's integral), 

- ['{^ + cos <^ (^ - l^}«d<^. 

When n is not an integer it is necessary to state which of the branches of 
the many-valued function in the integrand is to be taken: we shall take 
that branch of the function [z + cos <f){^ — 1)*}** which reduces to unity when 
taken by the process of continuation along a straight path to the point z^l. 
It will appear later that it is immaterial which branch of the two-valued 
function (^ — 1)* is taken. 

(A) Proof applicable only to ike Legendre polynomials. 

When n is a positive integer, the result can easily be obtained in the 
following way. We have 



2 A«P«(^)-(1-2A^H-A«)-*. 



But 



(i-2A.+A«)-*=ir^^f-^^ 



d<f> 



(l-A^)-A(-2»-l)*cos<^' 
as is seen by applying the ordinary formula for the integration of 

d<t> 



: 



a + 5cos<^* 
Expanding the integrand of the integral in ascending powers of h, we have 

(1 - 2A^ + A«)-* « - i A« ['{z + cos ^ (^«- 1)*}**^^, 

^»=o Jo 

and on equating coefficients of h^ on the two sides of this equation, the 
required result is obtained. 

As however the theorem is true whether n is an integer or not (ie. as it 
is equally true for the Legendre functions and the Legendre polynomials), 
it is necessary to have a general proof independent of the character of n ; 
this will now be given. 

(B) General proof 

First, we shall shew that Laplace's integral satisfies Legendre's equation. 



I^^p*^^^^^^" 



216 TRANSCENDENTAL FUNCHONa [CHAP. X. 

For if we write 

'fJo 



we have 



= - ['{^H- COS <^(^«- !)*}«-» {n8in»^-l-^C08^(^-l)-*}d^. 
But 

f '{-? + COS <^ (z« - !)*}«-« sin«<^d^ 

« — {^ + cos ^ (j8* - 1)*}*^ 8in<^ COS ^ 

4- Tcos A :^ [sin <^ [-? + cos <^ (i^ -!)*}«-•] d^ 

« / ' {^ + cos ^ (^ - 1 )*} »-« COS" <^ rf^ 

- (n - 2) f '{^ + COS <^ (z« - 1)*}'^» cos <^ (-?» - 1)* sia«^d<^ 
-'o 

«r{^ + cos^(^«-l)*}~-^ci<^~(n-l)r{xr + cos<^(-8«--l)*}»-^sin«<^d^ 
Jo Jo 



+ (n - 2) ^ rsm*^dif> {2: + cos ^ (-e* - 1)*}**"*. 
Jo 



Therefore 



n 1 {-ar + cos ^ (-«■ - 1)*}*^ sin«^(i<^ 
Jo 

=rf'{2r + cos^(-e»-l)*}'«d<^ + (n--2)2rr{ir + cos<^(-^«-l)*}~-«sin«<^^^ 
Jo Jo . 



Thus we have 



(l-^)g-2.| + n(„ + l)y 



= - (n - 2) « f ' {« + COB (^ («• - !)*}"-• sin'^d^ 
9r Jo 

- - « (^ - 1)-* f'fz + cos ^(«' - l)»}»-'cos <tKi4> 
= « !L ^ (^ _ 1)-* f ^ [U + cos ^ (^ - 1)*}"-* sin ^] # 

TT Jo w<P 

-0, 



119] LEGENDRE FUNCTIONS. 217 

which shews that Laplace's integral satisfies Legendre's equation, whatever 
n and z may be. 

1 -^z 
Now suppose that z is nearly unity, and put — ^— » u. Then the integral 

becomes 

1 r* 

-I {l-2w + cos^(-4w4-ti«)*J«d<^, 
which for small values of u can be expanded in the form 

1+- d<i 2 -^^ ^- — j^^ ^ {-2a + co8 A (-4w + w*)*K 

This is a series of powers of m*; the first terms (neglecting w*) are 

If* If' 

1 + 2inw* - I cos ^d,^ — %nu - / {1 + (n — 1) cos'^} d^, 

or 1 — inu ^ , 

2 

or 1 — w (w 4- 1) «. 

It is clear that odd powers of 'v^ can arise only in conjunction with odd 
powers of cos^ in the integrand, and so here vanish when integrated. 
Laplace's integral can therefore, when u is small, be expanded in ascending 
powers of u in the form 

1 — n (n + 1) w + OjW* + OjW* + a4W* + ... . 

But the coefficients a,, a,, ... can be found by substituting this expression 
in Legendre's equation, and equating to zero the coefficients of each power 
of u. We thus find that 

/ txy ^(^-~l)»"(^^y + 1)'(1 -l-n)...(r — 1 +n)(r-hn) 

r I r : 

and thus Laplace's integral is equal to 

jP(-n, n + l, 1, -^)» 

or (§118) to .PnW. 

We thus have, for all real or complex values of n, the result 

T^(z) « 1 ['{^ + cos <^ {z^ - 1)*}» d^. 

It must be observed that as the power-series Fi-^n^ « + l, 1, — g— j 
was used in the proof, this proof is valid only for values of z which satisfy 



218 TRANSCENDENTAL PUNCTrONS. [CHAP. X. 

1 1 — i^l 

the inequality - — ^ — < !• ^^ however Pn (z) is an analytic function of z, 

'the result will be true for a more extended region including this, provided 
the integral 

is an analytic function of z within this more extended region: since if 
these two expressions are equal for any region however small in which they 
are analytic functions, they must be always equal so long as they remain 
analytic functions. But it is easily seen that for the integral 



i J'{^ + cos <^ (^« - l)*j« d<^, 



every point on the imaginary axis in the -^-plane is a singularity: and 
therefore the region in the ^^-plane for which the equality 



"Jo 




is established is the region for which the real part of z is positive. 

Corollary. Since 
we have for all values of n, real or complex, the result 

Pn (Z) = - ['[Z + COS <^ (^ - l)»}-^» d<l>, 

TrjQ 
so long as the real part of ^ is positive. 

Example, If 
shew that 






(Binet) 



120. Tlie Mehler-Dirichlet definite integral for P« (z). 

Another expression for the Legendre function as a definite integral may 
be obtained in the following way. 

For all values of », we have by the preceding theorem 



P„ (^) = 1 Tf^r + cos ^ (^ -!)*}« d<^ 



In this integral, replace the variable <^ by a new variable h, defined by 
the equation 

A «= ^ + (^ — 1)*C08 ij> 



120] LEGENDRE FUNCTIONS. 219 

80 that 



and 



We thus have 



ctt«-(^-l)i8in<^d^, 
i(l-2A«: + A«)*-:(^-l)*8in<^. 



and therefore 



(1-2A^ + A»)* 



d<f>, 



Now write z = cos 0. Thus 






Pn(co8^)«- C^A'*(l-2Acos^ + A«)-*dA. 
Writing A « «•♦, this becomes 

^^^'^^^~^j.,(2cos<^-2cos^)*' 
or 

i> / ^\ 2 r^ cos (^ + i) i , , 

Pn (cos ^) » - I 7^7 ^T ^loi #. 

^ ' ^ J {2 (cos <^ — cos 0)}^ ^ 

This is known as Mehlers simplified form of Dirichlet's integral. The 
result is valid for all values of n. 

Example 1. Prove that, when n is a positive integer, 

*^ ' irj^ {2(cosd-co8<^)}* 



For we have 



Put 



r* dw IT 

Jo a+6-(a-6)co8ir*2a^*' 



a-(H-A)«, 6=l-2Ay+A«, 

The equation becomes 

IT /-y (i±.^)f^ 

Writing £«cos <^, y>«coB ^, this gives 

(l-2Acos^+A<)-*=- f'(l+A)8in<^(l-2Aco8<^+A2)(l+co8<^)-i(oo8^-co8<^)-*ci0. 

ir J 



Equating coefficients of A* on both sides, we have 

P,(cos^=^| ^«in(n+^)<^sin<^ 

{2 (cos ^- cos 0)}* 



^ ]. r* sin (nH 
I sm^coS"^ 

./ A 2 2 



220 



TRilNSCENDENTAL FUNCTIONS. 



[chap. X. 



or 



P»(co8^-- I {2 (006^-008 0)}-4 sin (n+i)<^. 
Example 2. Prove that 



PAoo.e).±.f^ 



2Aco8^+l)i 



dh. 



the integral being taken along a closed path which encloses the two points h^e*^^ and the 
conventional meaning being assigned to the radical. 

Hence (or otherwise) prove that, if ^ lie between }ir and }ir, 



2.4... 2n 



^-^^^^^-nzr^ik^) 



coa(nB+<t>) V C06(n^-h3<^) 

(2sintf)* ''■2(2«+3)~'(2sin^)l 

1«.3* oos(n^+6<^) 




where <t> denotes i^-^ir. 

Shew also that the first few terms of the series give an approximate value of F^{cob$) 
for all values of $ between and n which are not nearly equal to either or n. And explain 
how this theorem may be used to approximate to the roots of the equation P^ (cos ^=0. 

(Cambridge Mathematical Tripos, Part II, 1895.) 

121. Expansion of Pn(z) as a aeries of powers of - , 

z 

We now proceed to find an expansion of the Legendre function which is 
valid for large values of z. 

If the real part of 2^ be positive, we have for all values of n (fix>m 
Laplace's integral) 

^ Jo 
Now suppose that |-?| is very large : then this can be written in the form 

Expanding the integrand in ascending powers of - , this gives 

z 

We can evaluate 

/ (1 + cos <^)'* d<^ and | cos<^(l + co8<^)'*-*(i</> 
^0 Jo 






121, 122] LEGENDBE FUNCTION& 221 

by patting ^ » 2*^ and using the result 

and thus we find that P» {t) can be expressed by a series of powers of - , the 
first two terms of the expansion being given by the equation 

P .V 2»5"r(n+^) ( _ n(n-l) ) 

The general law of the coefficients in the series can without difficulty be 
found by substituting in Legendre's differential equation (§ 114) ; and iu this 
way we find that P» (z) can be eaypressed by the hypergeometric series 

P/^^ = 2VM[>^) /1-n n 1 1\ 

in the notation of § 14. 

This series has only been proved to hold when z is large and the real part 
of <g: is positive : but by § 14 it converges, and so represents an analytical 
function, over all the area outside the circle of centre and radius 1. The 
series therefore represents Pn(z) over this region. 

122. The Legendre functions of the second kind. 

Hitherto we have considered only one solution of the Legendre differential 
equation, namely Pn (z). We can now proceed to find a second solution. 

It appears from § 114, that the differential equation 

is satisfied by the integral 

^t* - l)"" (t - zy-' dt, 



/< 



taken round any contour such that the integrand resumes its initial value 
after making the circuit of it. Let D be a figure-of-eight contour in the 
^plane, enclosing the point t^ + 1 in one loop and the point t = — 1 in the 
other, and not enclosing the point t^z. Then after describing this contour, 
the above integrand clearly resumes its initial value, since it acquires the £su;tor 
e^ after describing the first loop, and this is destroyed by the &ctor e"**** 
acquired during the description of the second loop. D is therefore a possible 
contour. 



A 



222 TRANSCENDENTAL FUNCTIONS. [CHAP. X. 

A solution of Legendre's equation is therefore furnished by the function 
Qn {z)y if Qn {z) be defined by the equation 

it is supposed that, in describing D, the point t makes a positive, i.e. counter- 
clockwise, turn round the point ^ = — 1, and then a negative, i.e. clockwise, 
turn round the point ^ » + 1. The significance of the many- valued functions 
(^• — 1)" and {t^z)"^"^ will be supposed to be fixed in the same way as 
before. 

Another form of the integral may be obtained in the following way. 

Let the contour become so attenuated as to consist simply of a line 
joining the points — 1 and + 1, described twice, and two small circles round 
the points — 1 and + 1 : when the real part of (n -h 1) is positive, the parts 
of the integ^l arising from these two loops are at once seen to be infinitesi- 
mal; and thus we have 



= 2isin?i7rJ {l-fYit-zY^'^dt, 

«o Qn{z)^^,\\\-i^T{z-t)^'di, 

This last result is valid when the real part of (w + 1) is positive. When n 
is a positive integer, the original definition of Qn{z) becomes imdeterminate: 
in this case we can use the formulae just found. 

Qn{z) is called the Legendre function of the second kind and of order n. 

123. Expansion of Qn (z) as a power-series. 

We now proceed to express the Legendre function of the second kind as 

. . 1 
a power-senes m - . 

We have, when the real part of (71 -f- 1) is positive. 

Suppose that 1 ^ 1 > 1. Then the integral can be expanded in the form 

«.(«)-ss^ /la -«■)■ (1 -;)""' <» 



123] LEO£NDBE FUNCTIONS. 223 

as is seen on writing r for 28, since the integrals arising from odd values of r 
obviously vanish. 

Writing ^ « M, we can evaluate the coeflScients of powers of - as follows : 

z 

r(n + i)r_(* +i) 

r(n + « + f) "' 
and thus the formula for Qn {z) becomes 

Q (^\J'^^ r(n4-l) 1 y/n-hl 71 + 2 3 1\ 

VnW 2»^ir(n + f)^«+i^ V 2 ' 2 ' ^^r z'J' 

This is the expansion of the Legendre function of the second kind as a 

power-series in - , corresponding to the expansion obtained for Pn (-?) in § 121. 

z 

The proof given above applies only when the real part of (n + 1) is positive ; 
but a similar process can be applied to the integral 

^ ^ 4i sin riTT j j[) 2^ ^ / \ / 

the coefficients being evaluated in the same way as those which occurred in 

the expansion of the Legendre function Pn (^) in ascending powers of —5— ; 
the same result is reached, which shews that the formula 

O r^^ '^^ r(n-H) 1 r^/n-H n + 2 3 1\ 

VfiW*2«+ir(n + f);?«+^^ V 2 ' 2 ' ^"^2' W 

is true for all values of n, real or complex, and for all values of z represented 
by points outside the circle of centre and radius unity. 

Example 1. Shew that, when n is a positive integer, 

We can write Legendre's differential equation in the form 

(l-^g-2*|+n(«+l)«-0. 
It is easily verified that this equation can be derived from the equation 

(!-«») 5+2(»-l)«J+2«*=0, 

by differentiating n times and writing t£» ;^ • 



224 TRANSCENDENTAL FUNCTIONS. [CHAP. X* 

Now one solution of the latter equation is a7«(2'- 1)* ; and a second solution oan be 
derived bj the ordinary process for finding a second solution of a linear differential 
equation of the second order, of which one solution is known. Thus two independent 
solutions of this equation are found to be 



It follows that 



(««-l)» and (««-l)» I (t^-l)-«-irfr. 



is a solution of L^;endre's equation. As this expression, when expanded in ascending 

powers of - , commences with a term in d'^'^ it must be a constant multiple of Q^ (t) ; and 

on comparing the coefficient of «"*~* in tnis expression with the coefficient of z"^"^ in the 
expansion of Q^ (')» ^ found above, we obtain the required result. 

Example 2. Shew that, when n is a positive integer, the Legendre function of the 
second kind can be expressed bj the formula 

For on expanding the int^;rand (v^-l)"'"'^ in ascending powers of -, the right-hand 

V 

side of the equation takes the form 

^" ax- /.■'*>"■ {^' - SSI- '^^^m^ -..} . 

and on performing the integrations this becomes 

n! f 1 ■ (n-H)(7t+2) 1 1 

, (2» + l)(2»-l)...3.1 |«* + i"*" 2(2»+3) z^+s -«■—/» 

or §«(«). 

Example 3. Shew that, when n is a positive integer. 

This result can be obtained by applying the general integration-theorem 

to the preceding result. 

124. The recwrrence-formvlae for the Legendre function of the second 
kind. 

The functions PnW and Qn(^) bave been defined by integrals of pre- 
cisely the same form, namely 



/(t«-i)»(^-^)-«-id^. 



It follows therefore that the general proof of the recurrence-formulae for 
Pn(^)t given in § 117, is equally applicable to the function Qn(^)'i a^id hence 
that the Legendre function of the second kind satisfies the recurrence-fbrmtdae 



124, 125] LEGENDRE FUNCTIONS. 22& 

dQn(z) dQ^Az) 



^Z 



g^' = n(2^.(4 , 



dz 
nQn (z) - (2w - 1 ) zQ^, (z) + (n-l) Q^ (z) = 0, 

, dQn (Z) dQn-r (Z) _ ^^ , . 

dQn+1 JZ) dQn-i (Z) ,^_ . , , ^ . 

-rfi d^^ (2n+l)Q„(.). 

ft 

(^ - 1) ^^ = nzQ„ (z) - riQ^, (z). 

126. Laplace's integral for the Legendre function of the second kind. 
Consider the expression 



y = f {z + cosh (z* - 1)*}-**-^ d0, 
Jo 



in which z is supposed not to be a real negative number between — 1 and 
— 00, and the real part of (n + l) is supposed to be positive; under these 
conditions the integral certainly exists. 

If now we form the quantity 
(which occurs in Legendre's diflferential equation), we find for it the value 



- (n + ly i {z-k- (f - 1)* cosh ^}-^« sinh* Odd 
Jo 







+ (w + 1) f {2: + («» - 1)* cosh e]-"*-* dd 
Jo 

4- (n + 1) 2r («« - 1)-* f {^ + (^ - 1)* cosh ^j-"-' cosh dd0. 

Jo 

This expression can be transformed, by integration by parts, in exactly the 
same manner as the corresponding expression found in the discussion of 
Laplace's integral for Pn (z), in § 1 19 ; and thus it is found to be zero. The 
quantity y therefore satisfies Legendre's equation. 

In order to compare y with the solutions Pn (z) and Q» (z) which have 
already been found, we suppose that | ^ | is large, and write y in the form 



g-nr-i 



J*|l+cosh^(l-^ + ...)p 'd0, 



W. A. 15 



226 TRANSCENDENTAL FUNCTIONS. [CHAP. X. 

which when expanded as a power-series becomes 

^0 a^ a^ ch^ m 

where ao= f (1 + cosh ^)-«-i d^ 

^0 



/, 



'^,.v'(l-v)-*dv. where . = j-j-^. 



= ^— rr-8(^ + l> i)> where B is the Eulerian 



2n+i 



integral of the first kind, 






Now any expression of the form (1) which satisfies Legendre's differential 
equation must be a multiple of On(^) (since, by substituting the expansion in 
the differential equation, we can determine the coefficients ai,a^,a^,.,, uniquely 
in terms of a©, which shews that all expressions of the kind are multiples of 
any one of them); and as the value found for ao is equal to the coefficient of 
the initial term in the expansion of Qn (z\ we have 

y^Qnizy 

Thus we have the result 



Qn W = [ {^ + (^* - 1)* cosh ^}-*-i d0, 



which may be regarded as the analogue of the Laplace's integral already 
found (§119) for Pn(^). 

The theorem is valid only when the real part of (n + 1) is positive ; and 
the proof has assumed that | ar | > 1 ; but the equivalence of Q„ (z) and the 
integral, having been proved to subsist for this range of values of z, must 
continue to subsist for all values of z, continuous with this range, for which 
the integral continues to represent an analytic function of z ; and hence the 
theorem holds for all values of z except those which are real and less than 

— 1, which are singularities of the integral 

126. Relation between Pn(z) and Qn(z), when n is integral. 

When n is a positive integer, and z is not a real number between 1 and 

— 1, the functions Qn{z) and Pn(z) are connected by the relation 

which we shall now establish. 



126] LEGENDRE FUNCTIONS. 227 

When 1 ^ 1 > 1, we have 

l/>«.4^-U>w^('-!-^^••■)• 

Now if (n + A) is an odd integer, we have 

r -P" (y) ^^y = f '-P- (y) y^ - ?p- <y) ^^y = «. 

J -1 Jo Jo 

If n is less than ifc, and (n 4- A;) is an even integer, we have 

\ \\Pn (y) y'dy = l^n (y) y'dy 

1 fi d^ 
(by Rodrigues' theorem) = g^j J^ y* ^^n (y' - 1)" dy 

1 f^ 

(integrating by parts) = ^^ A; (A; - 1) . . . (Ar - w +1) y*"* (1 - ff dy 

JL Til J 

=2^!^'^*""^^^*""^)-(*-^-^^^^(^r^' ^ + i) 

- A?(A;-1)(Aj-2)... (Jk-n + 1) 
'"(fc + n+ l)(A; + ?i-l)...(A?-n+l)' 

If on the other hand h is less than n, and (n H- &) is an even integer, the 
same process shews that the integral vanishes. 

Therefore 

A;(A;-l)...(i-n4-l) 1 



2J.i '^^^^i^-y"" (A: + n4- 



1) (i 4. n - 1)... (A;-n+l) -e*+i ' 

where the summation is taken for the values A; = n, n 4- 2, 71 + 4, n + 6, ... 00 . 
But this expansion, by § 123, represents Q,n{s^- The theorem is thus 
established for the case in which |£:| > 1. Since each side of the equation 



Qn(.) = |/;P„(y);^ 



represents an analytic function even when \z\ is not greater than unity, 
provided z is not a real number between — 1 and 4- 1, it follows that, with 
this exception, the result is true universally. 

Example, Shew that Q« {z\ where n is a positive integer, is the coefficient of A* in the 
expansion of(l-2A«+A*)-ico8h-i j- ^ " \ . 

For 



n=o 11=0 * 7 -I *— y 

_1 n (l-2Ay+A«)~*d y 

=(1 -Steft+A«)-*co8h-> {-f J J J • 



16—2 



228 TRANSCENDENTAL FUNCTIONS. [CHAP. X. 

127. Development of the function (t^x)"^ as a series of Legendre 
polynomials in x. 

We shall now obtain an expansion which will serve as the basis of 
a general class of expansions involving Legendre functions. 

We have, by the recurrence-formulae, 

(2n + l)a:Pn(a:)-(n+l)Pn+i(a?)-7iP,^i(a:) = 0, 

(2n + 1) zPn {z) - (n 4- 1) Pn+i {z) - nP^i {z) - 0. 

Multiply the firat of these equations by Pn{^\ the second by Pn(^)> 
and subtract: we thus obtain 

(2n + l){z^x)Pn{z)Pn{x) 

= (n + 1) {Pn^, (Z) Pn {X) - Pn (z) Pn^i (x)} 

- n {Pn (Z) Pr^, (X) « Pn (x) P^^ (z)]. 

Write n = 0, 1, 2, 3, ... n successively, and add the resulting equations. 
This gives 

{P,(x)P,(z)-^SP^(x)P,(z)+...-\-(2n-\^l)Pn{x)Pn(z)}(z-x) 

= (n + 1) {Pn+i (Z) Pn (^) - Pn+1 (^) Pn (z)]. 

Divide throughout by (z — x){z — 1\ and integrate from -? = — 1 to 
^ = + 1. 

Thus 

x\'^\2r^\)Pr{x)^^dz 

^r., {.z-x)iz- i) ^^»^- <^> ^» ^"'^ ■ ^•^' ^''^ ^" ^^>' "^ 

(by partial fractions) = — — I -— — {Pn+i (-8^) Pn (^) - Pn+i (^) Pn (^)} d-^ 

"/I't^ {^»+i (^) ^» (^) - -Pn+i («') i^n (^)l d^ . 

Now by the result of the last article, the left-hand side of this equation 
can be written 

-2l(2r-hl)P,(a;)e,(0. 



In the first integral on the right-hand side, replace the integrand by its 

n 

value 2(2r H- l)Pr(a?) Pr{z), and integrate : only the first term survives, since 



I Pr{z)dz^O, 

when r is an integer greater than zero ; so the integral has the value 2. 



127] 



LEGENDRE FUNCTIONS. 



229 



We thus have 



2 (2r + 1) P, (w) Qr (t) = -L. + }±l {P„ (x) Qn+, (t) -P„+. (x) Q„ (0). 

This equation is valid for all values of n. Let us now see if x and t can 
be so chosen as to make the last part of the right-hand side tend to zero as 
n tends to infinity. We have, from Laplace's formulae for the functions 
Pn and Qn, 

p. w «.„ «) - p... (.) «. «) -If J] {i^i^r ^ #"*. 

where A denotes a quantity which is finite and independent of n. 

It is clear that this double-integral tends to zero only when, for all values 

of (f> between zero and tt, and all values of -^ between zero and infinity, 

the inequality 

a;-|-(a^«--l)*cos<^ 



t-h(e*-l)*coshi|r 



<1 



^ = K^^i 



t 



'Ih^i)' 



is satisfied. 
Writing 
the inequality becomes 

uA viu — I cos<f> 

u \ uj ^ 

The left-hand side of this relation has its maximum value when cos^s 1, 
the value being 2 1 u |. 

The right-hand side similarly has a minimum value equal to 2 { t; {. 

The condition thus becomes 



v^ h ft;r--jcosh'^ 



u\ < \v 



or 



|a:H-(^-l)*|<|t-h(<«-l)*|. 

This inequality shews that the point x must be in the interior of an ellipse, 
which passes through the point t, and which has the points H-l, — 1 for its foci : 
for if a be the major axis of this ellipse, then 

t = a cos + i (a* — 1)* sin 0, 

where is the eccentric angle of t in the ellipse ; and thus 

(t^ -. l)i = (a»-- 1)* cos ^-huisin 0, 

and e + (^» - 1 )* = {a + (a« - 1)*} e^, 

so that |^ + (^2-i)*| = a + (a»-l)*, 

and hence the above inequality shews that the semi-axis of the ellipse which 
passes through x is less than the semi-axis of the ellipse which passes through 
t, i.e. that x is within the ellipse which passes through t. 



230 TRANSCENDENTAL FUNCTIONS. [CHAP. X. 

Hence if the point x is in the interior of the ellipse which passes through the 
point t amd has the points + 1,-1, for its foci, then the expansion 

-i- = I (2« + l)P„(a:)<2„(0 

t — X n=0 

is valid, 

128. Neumann's theorem on the expansion of an arbitrary function in 
a series of Legendre polynomials. 

We proceed now to discuss the expansion of any arbitrarily given function 
in terms of the polynomials of Legendre. The expansion is of special interest, 
inasmuch as it represents the case which stands next in simplicity to Taylor's 
series, among expansions in series of polynomiala 

Let f{z) be any function, which is regular at all points in the interior of 
an ellipse C, whose foci are at the points -^ = — 1 and z^ + l. We shall 
shew that it is possible to expand f{z) in a series of the form 

aoPo(^) + a^Pi {z) 4- a^P^{z) + a^Pj,{z) + ..., 

where Oq, a,, a,... are independent of z : and that this expansion is valid for 
all points z in the interior of the ellipse G, 

For let £r = ^ be any point on the circumference of the ellipse. 

Then we have 

or /(^)= S anPn{z\ 

where an= gTri j /(OQn(0^- 

This is the required expansion. 

Another form for a^ can be obtained in the following way. 

We have 



_ 2n + 1 



rV/<^>^^--2/>^<^>.-^ 



2n + 1 /•+! 



2 
2n + 1 r+i 



r>'<^^^^\{-Bf 



f_f{y)Pn{y)dy. 



2 
The latter is the more usual form for On- 



128, 129] LBGENDRE FUNCTIONS. 231 

Example 1. Shew that the semi-axes of the ellipse, within which the series 



converges, are 



K''+D*"'*K''-J)' 



where p is the radius of convergence of the series 
Example 2. If 

Vy+l/ ' (^+l)(y-l) 

prove that 

129. The associated functions Pf!^{z) and Qn^iz)- 

We shall now introduce a more extended class of Legendre functions. 

If m be any positive integer, the quantities 

will be called the associated Legendre functions X>f the nth degree amd mth 
order, and will be denoted by Pn^(z) and Qn^(z) respectively. 

We shall first shew that the associated Legendre functions satisfy a 
differential equation analogotcs to the Legendre differential equation. 

For let the Legendre differential equation 

d^v 
be differentiated m times, and let v be written for -r-^ . 

We thus have for v the equation 

(l--?»)^,-2ir(m + l)^ + (n-m)(nH-m + l)t; = 0. 



m 



Write w = v(l-2:*)2; 

the equation becomes 

.- -. dhv e, dw ( , , . mM ^ 

This is the differential equation satisfied by the functions 

P„«»(^) and Q„'»(4 



232 TRANSCENDENTAL FUNCTIONS. [CHAP. X. 

Several expressions for the associated Legendre functions can be obtained 
easily from the above definitions. 

Thus from Schlafli's formula, we have 



m 



iiTr* Z J (J 

where is a simple contour enclosing the points ^^1 and t^z, but not 
enclosing the point ^ = — 1. 

From this result, or from Rodrigues' formula, we have, when n is 
a positive integer, 

130. The definite integrals of the associated Legendre fimctions. 

The theorems already given in § 115, relating to the definite integrals of 
the Legendre functions, can be generalised so as to be stated in the following 
form : When m and n are positive integers, 

j Pn'^{z)P/^(z)dz^O, when r<n, 

and r^P^''('^}'^'-A,?^r 

J-i^ ^ 2n + l(n— m)I 

To establish these results, we use the identity 
which gives 

L r (^ + >")' / nm 



{^(^'-i)*H^<^- '>"}''' 



(integrating by parts) = aS^^^J' jd^(^-l)f <i' 

2 (n + m)! 



2n+ 1 {n''m)V 



i 



130, 131] LEGENDRE FUNCTIONS. 233 

We can prove in the same way the other result stated, namely that 

I P„"* (z) Pr"" (js) dz = 0, when r + n. 

For this integral in the same manner reduces to a multiple of 

which is zero when n and r are diflFerent. 

131. Expression of Pn^{z) as a definite integral of Laplace* s type. 

The associated Legendre functions can be expressed by means of definite 
integrals of the same type as those found in § 119 and § 125, as will appear 
irom the following investigation. 

We have 

/ {z + cos (^ - !)*}«-*" sin^ (f>d<l> 
Jo 

= - {2r4-cos0(^>-l)*}**-^sin*^-^<^cos<^ 

+ 1 COS (f>-j-r [sin««-i (f>{z + cos <^ {z^ - 1)*}**-*^] d<f> 
Jo a<f> 

= (2m - 1) f 'cos« <f> sin«^-^ <l>{z+ cos (z^ - l)*}**"^ d(f> 

Jo 

- (n - m) ('{z + cos (z^ - l)*}*-«-i cos <^ (z^ - 1)* sin«~ <l>d<f> 

Jo 

= (2m - 1) f 'sin**^ (f>{z-\- cos ^ (z^ - l)l}»-«» d<^ 

- (2771 -1)1' sin»^ { J + cos <^ (^ - 1 )*}«-^ d^ 

- (71 - m) / (^ H- cos <^ (z^ - 1)*}*-^ sin«'~ <^d0 

+ (n-m)z I f^r + cos {z^ - l)*}»-«-i 8in»» ^d^. 
Jo 

We thus have 
(n + m) j {z + cos <^ (^ - l)i}»-»» sin*» <^d<^ 

= (2m - 1) Tsin^^-^ <f>{z + cos <^ (-g« - 1)*}*»-^ d<^ 
Jo 

- ^^ ^ I {^ + cos (f> {f - l)*}~-« 8in»^^ <^1 

+ ^^ I {-g: + cos <^ (-?» - l)*}»-*»(2m -1) 8in*^»<^ cos ^d<^, 



234 TRANSCENDENTAL FUNCTIONS. [CHAP. X* 



"^ 2m 



?:±^ ['[z + cos A (^ - 1)*}'-^ sm«~ <^d<^ 
Zm — 1 J 



= rsin««-« <^ {ir + cos (z* - !)*}*-« {1 + r (-r« - 1)-* cos <^} d<f> 
Jo 

= — — - -J- \{z + COS 6 (z^ - l)i}«-"»+i 8in«*-» <bd6. 
n — m + 1 d^^Jo T-v /) Y' r 

Thus if we write 

Tm-l (2: + cos (^ - !)*}'*-« sin** <^c^, 



we have /„=_-(2^1) %> , 

(n + m)(n— OT+1) dz 

and therefore I - (2m- l)(2m-3) ... 1 ^ 

** (n + m)(n + m-l)...(n-m + l) cJ^** 

But /o = f '{^ + cos <^ (^» - 1 )*}*» d<^ = irPn {z\ 

Jo 

when the real part of z is positive. 

Therefore /. = , (2m -1) (2m -3) 1.^ ,, ;^Pn(.) 



= (2m-l)(2m-3)...1.7r ^ -^ ^ 

(n + m)(nH-m-l)... (n-m + l)^ '^ "^ ^ ^' 

"""^ ^~ ^^^^ (2m-l)(2m-3)...1.7r ^^"^^ 

X I {-^ + COS <^ (^« - 1)*}»-^ sin*** ^ d^. 
Jo 

This result expresses Pn^(z) as a definite integral of Laplace's type, valid 
for all values of n when the real part of ir is positive. 

132. Alternative expression of PfJ^ (z) as a definite integral of Laplace* 8 
type. 

The formula last found can be replaced by another result, found in the 
following way. 

If in Jacobi's well-known theorem* 

Jy (co8^)co8m^d^ = 1.3.5 ..\2m-l) /o'-^'"'^^^"*>°^°"'^'^- 
we take /(cos ^) = {^ + (z^ — 1)* cos <f>}^, 

* CreUe's Journal^ xv. 



H 



132, 133] LEGENDRE FUNCTIONS. 235 

80 that 

/w (cos <^) = n (n - 1) ... (n -m + 1) (z^ - 1)^ {2:+ (^ - 1)» cos <^}"^, 
we obtain 

I {^ + (^— l)*cos^}**cosm^d^ 
Jo 

^ njn-l) ... (n-m-hl) , _ , .f 
1.3.5...(2m-l) ^ ^ 

X I {z+ (z^- 1)* cos 0}»*-^ sin^ <f)d<f> 
Jo 



(n + m)(n +m— 1) ... (/i + 1) 
Therefore 
p^m /^\ ^ (^ + m)(n4-m-l) ... (n + l) , ^vf 

TT 

X 1 {2^ + (^ — 1)* cos (^J** cos m<f>d(f>. 
Jo 

This formula is valid for all values of w, and for all values of z whose real 
part is positive ; m being a positive integer. 

133. The function Cn*" («). 

A function connected with the associated Legendre functions F^^ {z) is the function 
On*" (^)) which for integral values of p is defined to be the coefficient of A** in the expansion, 
in ascending powers of A, of the quantity 

(l-2A^+A«)-»'. 

It is easily seen that C^*" (z) satisfies the differential equation 

<Py (2v + l)zdy n(n + Qv) 
d^^ z^-\ dz z^-l ^""* 

For all values of n and y, it may be shewn that C^ {z) can be defined by a contour- 
integral of the form 

Constantx (1 -««)*- ^ f ^lzf)-l^dt. 

When n is integral, we have 

Cy(z^= (-2)*v(i^4-l)...(v4-n-l) 

* ^ ^ n! (27i+2v-l)(2»+2v-2)...(n+2i^) 

which corresponds to Rodrigues' formula for P^ (z) ; in fact, since 

F^(z) = Cj{z\ 
Bodrigues' formula is a particular case of this formula. 



236 TRANSCENDENTAL FUNCTIONS. [CHAP. X. 

When r is an integer, we have 

«-'*^^^"(2r-l)(2r-3)...3.1 (£2^ *^^^' 
whence we have 

The last equation gives the connexion between the functions C^^ (z) and P/ (2). 

This function C/ (z) has the following further properties, analogous to the recurrence- 
formulae, 

<t;(.)-c.(^)-£c;w=o. 



Ow-<l«="-:^c'/(.). 



r-i 



Miscellaneous Examples. 

1« Shew that when n is a positive integer, 

p^{z)J-z:^-p(u^+z^r^, 

where t^* is to be replaced by (1 -^) after the differentiation has been performed. 

2. Prove that when w is a positive integer, 

(Cambridge Mathematical Tripos, Part I, 1898.) 

3. Shew that 

• -1 w « fl.3.6...(2w-l)l«,„ .. „ ... 

(Catalan.) 

4. Prove that 

is zero unless m- n«i ± 1, and determine its value in these cases. 

(Cambridge Mathematical Tripos, Part 1, 1896.) 

6. Shew (by induction or otherwise) that when n is a positive integer, 

(2»+l)rP,*(«)(fc-l-*P,«-2i(P,«+i','+"+^»-t)+2(A'P»+^»^»+-+^»-i^«) 

(Cambridge Matbematdcal Tripos, Part 1, 1899.) 



/- 



6. Shew that, if i{r is an odd number, 

1 



— jfe-2a»P»(4 

(1-22A+A«)« 



MISC. Exa] 



LKOENDRE FUNCTIONa 



237 



where 



8 a \i(t-») 



_A« 2^*-»(2n+l) / 8 3\ 

" (l-A«)*-«1.3.6...(it-2)V 8a? 8j^; 



^ ^-i(2i»-t+4)yi (2n+Jfc-2)^ 



where a: and y are to be replaced by unity after the differentiations' have been performed. 

(Routh.) 



7. If 



n-O 



shew that 



and 
and 

where 



2 (n+1) 72»+i -3 (2n+l) /?^+(2n- 1) i2»_2=0 
4(4a3-l)/J^'"+96««72»"-«(12n«+24w-91)i2^'-w(2n+3)(2n+9)i2»-0, 



K-J^ , etc 



(Pincherle.) 



8. If m and n be positive integers, and m^n, shew that 



P (z)P (z)== 2 A„,^rArA^.r ( 2n+2m-4r+l\ 



where 



^«' 



1.3.5... (2m- 1) 



ml 



(Adams.) 



9. Shew that P^ (z) can be expressed as a determinant in which all elements parallel 
to the auxiliary diagonal are equal (i.e. all elements are equal for which the sum of the 
row-index and column-index is the same) ; the determinant containing (2n- 1) rows, and 
its first row being 

(Heun.) 



_1 1 11 

^ 3' 3''-"5' 5^'"*2w-l 



t2. 



10. Shew that 



11. Shew that 



2 r {z(l-i^-2t(l^m^ ^^^ 



12. Shew that, when n is a positive integer, 



§^(cos^) (-1)** 8» fl 



f^tn-*-! 



n! 8«* 



(2V*«(r-+i)}' 



where ««=r cos ^. 



(SUva.) 



(Catalan.) 



238 TRANSCENDENTAL FUNCTIONS. [CHAP. X 

13. Shew that the complete solution of the Legendre differential equation is 



y=^p.(.)-H5P.w/;^3-^^^,. 



14. Shew that 



{2 + (2i^l)i]a^ 2 5«§2m.«.lW, 

where 

„ o(a+2 w+i) r(m~^)r(m-a-^) 

"*" 2tr ml r(m-a+l) * 



•'x+(««-l] 



dh. 



15. Shew that, when the real part of (w+ 1) is positive, 

and 

/-^-(^t-i)* A** 

16. Prove that 

(Cambridge Mathematical Tripos, Part II, 1894.) 

17. Shew that, if n be a positive integer, 

18. Shew that 
and 

where n is a positive integer, and z > 1, and where log ^ is to be changed into log 

z— 1 \ —z 

if « is numerically less than unity. 

Prove also that 

V 2 3/ P223« V 2 / ■*"••* 



whereit=l + - + 5+...+-. 
2 3 n 



(Cambridge Mathematical Tripos, Part II, 1898.) 



MISC. EXa] LEGENDRE FUNCTIONS. 239 

19. Shew that 

-P«*»(«)=-^ ^-^, ^f^F{n, n+m, m+1, s^). 

20. Prove that, if 

^•"n (n«-l)(n2- 4) ...{»«-(«- 1)2} (w+«)^^ ^^ ^» 
then 

,. -P 2^2n + l) 2n+3 

3 (271+3) p , 3(2n+5) ( 2n+3)(2n+ 5) 

y3"^-+8 2«-l ^»+i"^ 271-3 *-i (2w-l)(2w-3)^'-5' 

and find the general formula. 

(Cambridge Mathematical Tripos, Part II, 1896.) 



21. If 



shew that 



22. If 



On"" {^^1 - (^ - 1)* (^1^ - 1)* cos <^} 

_ n(2y-2) J^-* 4An(n-X){n(v+X-l)}g 

{n(y-l)}2xfo^ ^^ n(n+2i.+X-l) 

(Qegenbauer.) 



<^» («) = I **(^- 3^2 + l)-i i»c?<, 

where ^i is the least root of ^— 3te+ 1 =0, shew that 

(27i+l)crn+i-3(2w-l)z(r„.,+2(n-l)(rn-2=0, 
and 

4(4«5-l)(r,,'"+1442j<r,»"-«(12w2-24n-291) <r,»'- (n-3) (2n-7) (2n+5) cr,»=0, 
where 



o-n' = - ^} , etc. (Pincherle.) 



/'/ 



23. Shew that 



(Hobson.) 



g^n>(,)^^>i r(.^+l) r COShj^t. 

r(n-w + l);o {r+(22-i)icoshw}-+i 
where the real part of (n+l) ia greater than m. 

24. The equation of a nearly spherical surface of revolution is 

r=l+a{P,(cos^)+P3(co8^ + ... + P2n-i(cos^)}, 

where a is small ; shew that to the first order of a the radius of curvature of the 
meridian is 

l+a 2 {n(4wi+3)-(r?i+l)(8m + 3)}P8,n+i(cos^). 

(Cambridge Mathematical Tripos, Part 1, 1894) 



CHAPTER XL 



Hypbrgeometric Functions. 

134. The Hypergeometric Series. 

We have already in § 14 cjonsidered the hypergeometric series* 

a.b a(a + l)b{b + l) a(a + l)(a + 2)b(b + l)(b + 2) 
■^l.c "^ 1.2.c(c+l) "^ 1.2.3.c(c-rl)(c + 2) "^ "' 

from the point of view of its convergence. It was there shewn that the 
series is absolutely convergent for all values of z represented by points in 
the interior of the circle whose centre is at the origin and whose radius is 
unity. It follows from §22 that all the series which can be derived from 
the hypergeometric series by diflferentiation and integration are likewise 
absolutely convergent within the same region : and by § 55, the convergence 
is not only absolute but uniform over the interior of the circle, and the 
sums of the series obtained by differentiation and integration of the series 
term by term are the derivates and integrals respectively of the sum of the 
series. The hypergeometric series, together with the series which can be 
derived from it by the process of continuation (§ 41), will therefore represent 
an analytic function of the variable z; this function will be denoted by 
F(a,b,CtZ), 

Many of the most important functions of Analysis can be expressed by 
means of the hypergeometric series. Thus it is easily seen that 

(l+zY^F(^n,/3,/3,^zl 
log(l+5) = zi^(l,l,2,-5), 



6^ = Limiti^U,/8,l,j), 



and we have shewn in the preceding chapter that the Legendre functions 
may be represented by the series 

• The same was given by Wallis in 1655. 



134, 135] HTPBRGEOMETRIC FUNCTIONS. 241 

2-z-r(n-^i) (l^n n 1 1\ 

^«W- ^4r(n + l) ^\ 2 ' ""2' 2"^"' J«;' 

O /.,x_ 7r>r(n + l) 1 y/n+1 7i + 2 3 1\ 

These examples are sufficient to shew that the functions represented by 
the hypergeometric series are in some cases one-valued and in other cases 
many-valued. 

Example, Shew that 

^^F(a, 6, c, «)=^F(a + l, 6+1, c+1, z). 

136. FaZt^ o/* ^A« series F(a, 6, c, 1). 

We have shewn in § 14 that the series F(a, b,c,l) converges absolutely 
so long as the real part of c — a — 6 is positive. Suppose this condition to 
be satisfied. Then we have 



F(nhr^\-l. Tia + n)T(b + n)r(c) 
F (a, 6. c, 1) ^ Z^ ^uT(c + n)r(a)r(6) 



Tic) I 1 T{n + a)T{n-\-h)Vic-h) 

r(a)r(6)r(c-6),tonI r(n + c) 

r(c) I ir(» + o)5(n + 6,c-6)L 



r(o)r(6)r(c-6),.on! 

r(c) 



= r(a)Tl)l(c-b) iy'^''^IS^' - ^y-'-^^'dz. 
Writing z=^l — t, this becomes 

(writing xt^s) 



r(c) 



r(a)r(6)r(c-6);o Jo 
r(c) 



zrr\j '^1 e-'8^^if>-<'-<>-'(l-tf-\dt 



r(a)r(6)r(c-6) 



r(o)5(c-a-6,6) 



r(c)r(c-o-6) 
~r(c-6)r(c-o)* 

W. A. 16 



242 TRANSCENDENTAL FUNCTIONS. [CHAP. XL 

The hjrpergeometric series with argument unity can thus be expressed in 
terms of Oamma-functions. 

136. The differential equation satisfied by the hypergeametric series. 

The function represented by the hypergeometric series y='F(a,b,c,z) 
satisfies the differential equation 

'i^-l)2^ + l-o + ia + b + l)z]^ + ab!, = 0; 

for if the series be substituted for y in the left-hand side of the equation, 
the coefficient of 2^ is 

a(a + l)...{a + r-l)b(b + l)...(b + r-l) 
1 .2 ...r.c(c + l)...(c + r) 

{r(r'-l)(c-{-r)'-r(a-\-r)(b+r) -c{a+r)(b + r) + r(c + r)(a + 6 + 1) + oft (c + r)} 
or zero ; which establishes the result. 

Example, Shew that one integral of the equation 



IS 

where 



^F{m-/i, m-v, m-n+1, «), 
a-l--Oi+ir), 

6+l=m+n, 

137. The differential equation of the general hypergeometric Jimction. 

The differential equation found in the preceding article is a case of a 
more general differential equation, which may be written 



d2l' 



\ z — a z — b z — c ] dz 



{ aa'{a^b){a-c) ^ /3^(b^a)(b^c) ^ yy-(c-a)(c-6) j y ^^ 

( 2r — a z — b z — c J (^— a)(2r--6)(2r— c) 

* ...(A), 

in which a, 6, c, a, ^, 7, a , ff, y are any constants such that the equation 

a + /3 + 7 + a' + )S' + 7' = l 

is satisfied. This will be called the differential equation of the general hyper- 
geometric function. The form here given is due to Papperitz*. 

* Math, Annaleuj zxv. 



136, 137] HYPEEGEOMETRIC FUNCTIONS. 248 

We shall now shew that the differential equation satisfied by the hyper- 
geometric series is a particular case of this equation. 

For in the equation (A), write 

= 0, 6=00, c = l. 
The equation becomes 






y . f l-g-g' . I-7-7 ) dy ( ««' Tx' . oo/l _y_ n 



In this equation, let a and 7 be replaced by zero. We thus have 

d2^ \ z z—l J dz ziz—l) 

and in this equation the constants ol, y, fi, ^, are to be such as to satisfy 
the relation 

/g + a' + ZS' + y^l. 

This differential equation can be identified with the equation 
z(z-l)^ + {-c + (a + b + l)z]^ + aby = 0, 

which is the differential equation satisfied by the hypergeometric series, by 

writing 

y3 = a, )8'=:6, a' = l-c; 

which in virtue of the above relation gives 7' = c — a — 6. The differential 
equation of the hypergeometric series is therefore a special case of 
equation (A). 

We shall denote any solution of the general differential equation (A) by 

the symbol 

{a b c 

a ^ y z}. 

a! ^ 7' 

This notation is due to Riemann* ; it enables us to express our result thus : 

The hypergeometric aeries 

F(a, 6, c, z) 

is a solution of the differential equation of the class offu/nctions 



t 



00 1 



P a z 

[ 1— c b c—a—b 

* Abhandlungen d, K, Getell, d, WUtemchaften zu OdtHngen^ Tn. (1857). 

16—2 



244 TRANSCENDENTAL FUNCTIONS. [CHAP. XI. 

Although the hypergeometric series itself satisfies only a particular form 
of the differential equation (A), it is nevertheless possible to satisfy the 
general equation (A) by means of a function derived from the hjrper- 
geometric function. For by the transformation 

a? (5 — 6) (c — a) = (^r — a) (c — b), 

the differential equation (A) is reduced to the form 

cte* I X . a?— IJow? ( X a? — 1 } x{x — l) 

In this we take a new dependent variable, defined by the equation 

y = x*{l'-x)y u. 
The equation becomes 






, /I -a' + a , 1 -7' + 7\ du . ,^^ . x/i ^ /\> ^ /v 



Now the equation 

a + /S + 7 + a' + ^' + 7'sxl 

will be satisfied if j3, a\ ffy 7', are expressible in terms of three new constants^ 
a, 6, c, defined by the formulae 

/ /8 = a — a — 7, 
a' = 1 — c + a, 
^ = 6-a-7, 
7'=sc — a — 6 + 7. 

The differential equation for u can now be written 

d^u du 

x(x- l);i-i + {(1 +a + 6)a?-c}-Y- + a6M=0. 

But this is the differential equation satisfied by the hypergeometric 
series, a solution of it being 

F (a, 6, c, x). 

Hence we have, as one solution of the equation, 

u^F(a + /3 + y, a + /9' + 7, 1 + a - < a:), 

or y^x*(l- x)yF(a + ;8 + 7, a + /S' + 7, 1 + a - a', x), 

or, disregarding a constant factor, 



1 



• 138] HTPERGEOMETRIC FUNCTIONS. 245 

This is therefore a solution, expressed by a hypergeometric series, of the 
differential equation which defines the class of functions 

Ia b c 
a ^ y z 

The advantage of the differential equation (A) over the equation found 
in § 1 36, which is satisfied by the hypergeometric series, lies in its greater 
symmetry and generality. The points 2r = a, -e = 6, and z=^c, are called the 
singularities of the differential equation (A); the quantities a and a' are 
called the exponents at the singularity a; and similarly /8 and ^ are the 
exponents at h, and 7 and y' are the exponents at c. 

Example, Shew that 

00 1 ^ / -1 00 1 

fi y zn^p\ y 2/3 y 

k »' y' ] I y s/y y 

(Riemann.) 

This relation follows from the fact that the differential equation corresponding to either 
of the P-fiinctions is 

138. The Legendre functions as a particular case of the hypergeometric 
function. 

The expressions which have been found for Pn(^) and Qn(^) as hjrper- 
geometric series naturally lead us to suppose that Legendre's differential 
equation is a special case of the differential equation which defines the 
general hypergeometric function. That this is the case appears from the 
following investigation. 

If in equation (A) of the last article we take 

a = — 1, 6 = 00, c = l, 
we obtain the differential equation 

d»y ( l.,a^g^ 1-ry^ryO dy r 2ga^ 277M y 

d?^[ 5 + 1 ^ 5-1 ]dz^\ 5+l'^^^"^5-l](5-l)(5 + l) ''• 

If now in this equation we take a = 0, a' = 0, 7 = 0, ♦/ = 0, yS = n + 1, 
fi'ss^n, we obtain 

or (l-^')§-25^ + n(n + l)y = 0, 

which is the Legendre differential equation. 



246 



TRANSCENDENTAL FUNCTIONS. 



[chap. XI. 



It follows from this that any solution of Legendre's equation is a hyper- 
geometric function of the type 

-1 00 1 

p\ n + 1 z 
-n 

In the same way it can be shewn that the associated Legendre 
functions Pn^{z) and Qn^{z) are hypergeometric functions of the type 



P\ 



-1 



00 



2 " + 1 



1 

m 



\ 



I 



m m 

"2 -" "2 



Example 1. Shew that 



S^-«=^ 



-1 00 

— r w+r+1 
-n^r 



1 \ 

-r z 




• • 



Example 2. If «^si7, shew that the Legendre difierential equation takes the form 

c&,«"^\2,,"l-i7Jrfi7"^ 47(1-1;) 
Shew that this is a hypergeometric differential equation. 



139. TransformcUioris of the general hypergeometric function. 

We shall next consider the effect of performing certain transformations 
in connexion with the general hypergeometric function 

^a b c 




The differential equation satisfied by this function is 
d^ \ z^a z — b z — c)dz\ ~ ' 



z — a 



P^{b^a){b^c) . 77'(c-a)(c-6) 



z^b 



z — c 



\ y_ 

){z- a)(z - 



b)(z - c) 



= 0. 



In this equation, let the dependent variable be changed by the trans- 
formation 






139] 



HTPERQSOMETRIC FUNOTIONS. 



247 



\ 



The differential equation for y' is found after a slight reduction to be 

<fy ^ f l-«-«'-28 ^ l-/3-/y + 28 1-y-y' ) dy" 
dz* \ z — a z — b z — c)dz 



^ f (a + 8)(a' + 8)(a 
1 z ^a 



- 6)(a - c) ^ (/3 - 8)(/y - 3)(6 - c)(6 - a) 



+ 77 



,(c~a)(c-6) 






z — c I (-P — a) (^ — b)(z — c) 



^-6 



= 0. 



This is the differential equation of a hypergeometric function which has 
exponents a + S, a' + S, at the singularity a, and exponents )8 — S, )8' — S', at 
the singularity 6 ; and so we have 



C-5 



-./T.\« 



a" 




= P- 



a 6 c 

a + S )8— S 7 z 

a' + S /y-s y 



and hence in general we shall have 



(-;)'&:)■' I 



a b c . 
a ff 7 



= P 



a 



a + S id- 
a' + S /3' 



b c 

-S— € 7+6 
— S — € 7' + € 



It will be observed that by this transformation the exponent-differences 
a — a', )8 — )8', 7 — 7' are unaltered. 

Consider now the effect of transformations of the indepeAdent variable s. 

If we introduce in place of jer a new variable z\ defined by the equation 



5= 



Oi^ + ^i 
C^ + di' 



where Oifb^ Ci, di are constants, so that 



we have 



, - di^r + 6, 
z = 



and 



dy _ Oidi - 61C1 dy ^ (c^z + d^ydy 
dz (CiZ — Oi)* dz' Oidi — 6iCi de' 

rf^*^ ((hz-ihT dz {CiZ-OiY d/* 
^ 2c(c,z' + d,y dy {ci^ + d^Y d^ 



248 



TRANSCENDENTAL FUNCTIONS. 



[chap. XL 



Hence if we define quantities a', b\ c by the relations 

__ _ _ OiC + 61 

Cia' + di * Cjjb' + di * 

so that 






the general hypergeometric differential equation becomes 






[9. . (l-a-aO(Cxa^ + d,) , (1 -/3 - ^yXc^ + d,) 



z — a 



+ 



a-7"-70( 



77'(o'"-a')(c'-6')) 






y 



7T7T7-:7x=0. 



The coeflBcient of -7^ in this equation can be written in the form 



l-.a-a^ l-/8~)y 1-7- 7 , 1 
which, in virtue of the relation 



2ci-(l-a-a')Ci \ 



reduces to 



a + a' + /3 + )8' + 7 + y = l, 



i^g^g- i^ff^/y i-7^y 



Hence the differential equation reduces to the differential equation of 
the function 

a c 

p\a fi y z 

a' /3' 7' 






and thus we have the relation 



a 6 c 
P-la fi y z 
a' ^ 7' 



o 



=P- 



6' c' 

^ 7 ^' 
^ 1' 



This shews that the general hypergeometric function is unaltered if the 
quantities a, b, c, z are replaced by qualities a', b\ c\ z^, which are derived 
from them by the sam^e homographic transformation. 



140] HYPERGEOMETRIC FUNCTIONS. 249 

140. Ths twenty-four particular solutions of the hypergeometric differential 
equation. 

We have seen in § 137 that a particular solution of the general hyper- 
geometric differential equation is 



<z 



^r C-^)V f ./..,... /r ^ ,-, 1 .. - .'. I^^fe^j)} . 



We shall suppose that no one of the exponent-differences a — afy — ^, 
7 — 7' is zero : it is shewn in treatises on Linear Differential Equations that 
when this exceptional case occurs, the general solution of the differential 
equation involves logarithmic terms; the formulae will be found in a 
memoir* by Lindelof, to which the reader is referred. 

Now if a be interchanged with a', or 7 with 7', in this expression, it must 
still satisfy the differential equation, since the latter would be unaffected by 
this change. We thus obtain altogether four expressions for which 

(c — 6) (^ — a) 
(c — a) (-8^ — b) 

is the argument of the hjrpergeometric series, namely 

.fz-ay' /z-c\y „{ , r, , OP / -, . / (c - 6) (-gr - a)' 

these are all solutions of the differential equation. 

Moreover, the differential equation is unaltered if the quantities a, a', a 
are interchanged respectively with ^, ff, 6, or with 7, 7', c. If therefore we 
make such changes in the above solutions, they will still be solutions of the 
differential equation. 

Let a change in which (a, a', a) are interchanged with ()8, /S', 6) be 
denoted for example by 

/a, 6, c\ 

U, a, 0) ' 

each singularity in the bracket being interchanged with the singularity 
above or below it. Then there are five such changes possible, namely, 

(a 6 c\ fa h c\ (a h c\ (ah o\ (a b c\ 
\b c a)* \c a bj* [a c b) ' [c b a)' \b a cj' 

* Acta Soe, Scient. Fennieas, xix. (1898). 



250 



TRANSCENDENTAL FUNCTIONS. 



[chap. XL 



To each such change correspond (by interchanging a with a, etc. as already 
explained) four new solutions of the differential equation. We thus obtain 
twenty new solutions, which with the original four make altogether twenty- 
four particular solutions of the h3rpergeometric differential equation, in the 
fqrm of hypergeometric series. 

The twenty new solutions may be written down as follows : 

-c)j 









+y8-)8', 



+ /3'-/9, 



+ /3-/9', 



+ /8'-/8. 



+ 7-7', 



(a - h) (z 
{a-c){z-h)\ 



(a-b)(z-oX 
(a -c)(z — b) 



-6)1 
-c)\ 



(a - b) (z 

(O — C) (^r — 6) 

(6- 
(6 



- o) (g - c) | 

- c) (^ - a)j 

+ V-~ (^-»)(^-g) t 
+ 7 7. (6_c)(^_a)[ 

(6 — o) (« — c) 



+ 7-7'. 



+ 7'-7» 



(6 - c) (^ - o), 

(6 — a)(^ — c) 
(fc_c)(^-aX 



„_„' (fe-c)(^-a )) 
"' (6-a)(^-c)J 

(6-c)(ir-a) ] 
' (6-a)(^-c)J 

Q>-c){z-a) \ 
(6-a)(^-c)[ 

(6-c)(^-o) ] 
(6 — a) (5 — c) 

(a-c)(-? 

(a-6) (£- 
Xa — c^iz 

+ 'y "y' {fl-c){z-b)\ 

(a-6)(g-c)) 
-6)f 



+ a-a', 



+ a'-a, 



+ 7-7'. 



+ 7' -7. 



-6)1 



+ y'-% 



(a — c)(z 



141] HYPERGEOMETRIC FUNCTIONS. 251 

The existence of these twenty-four values was first shewn by Kummer* 

Example. Find the twenty-four solutions of the L^endre differential equation, 
corresponding to the above set of solutions of the hypergeometric differential equation ; 
and express each of them in terms of the two independent solutions P^ {z) and Q^ (z), 

141. Relations between the particular solutione of the hypergeometric 
diferential equation. 

Since the twenty-four expressions found in the last article are solutions 
of the same linear differential equation of the second order, any three of them 
must be connected by a linear relation with constant coefficients. 

We proceed to find the relations which thus connect them. 

First, consider the set of four solutions 

Vi* Vii Viz* yi6> 

it is clear that, in the neighbourhood of the point z=^a, each of them can be 
expanded in a power-series of the form 

4 (2r - a)* {1 -h £ (^r - a) + (7(z - a)» + ...}. 

But there is only one series of the form 

(z-a)* {1 -h£ (^r- a)-f C(2r- a)» + ...} 

which satisfies the differential equation; for the coefficients 5, C, ... can be 
uniquely determined by actual substitution in the differential equation. Let 
this solution be denoted by P<*>. 

Thus the solutions 

t/u Vty yi8, yi5 
must be mere multiples of P^*'. Moreover, 

for y, the factor A \s (a - c)y {a - 5)-c+y> ; 

for y, it is (a - c)y\a - 5)-<«+y'» ; 

for y^ it is (a — 5)^ (a — c)~*""^ ; 

and for y„ it is (a — bY{a — c)~*~^'. 

* Crelle*s Journal^ xv. 



252 TRANSCENDENTAL FUNCTIONS. [CHAP. XI. 

Thus we have 



^ ^ Ka — cl xa — hl 



xi-ja + ZS-K,. « + ^' + y. l-,„-a'.J--|(i^)} 



\a — c' \a — 



- 6\-"-r' 



6, 



x^{«+^+y, „+^+,. !+«-«, ;::^;;;:g } 



=(-«)• (src^D 



xP|a + ^ + -y. a + /9+r. l + a-«. (^ _ a) (^ - c) , 



■?■ fz-h\l>' 



=<->-c^r(^^) 



x'h^*''. ■>+«+»', 1+.-.', fti:^';::} }- 

Similarly solutions P<*\ P<^», P<^\ P^y\ P<y'> exist, each of which is equi- 
valent to four of the above hypergeometric series. 

Having thus classified the twenty-four solutions into six distinct solutions, 
namely 

p(a)^ p(e')^ pO)^ p(/r)^ p(y)^ p(y')^ 

we proceed to find the relations between these latter six solutions. We know 

that P'*> must be expressible linearly in terms of P<y> and P^y'K Let the 

relation between them be 

P(«) = a^p(y> + Oy^pty). 

We have then to find the coe6Bcients Oy and oy. 

Now this equation can be written in the form 



r{'*»^i. '*g*i: '+«-«'. I^^Jl 



z - 5\-»-r 



-('-)' (j^ro 



f |.^^^„ ^,^^„ ,^,.,., («_^|g^i 



5-6N— *' 






142] HYPERGEOMETRIC FUNCTIONS. 253 

Dividing throughout by the common factor (z — a)*, and writing ^ = a and 

z=sc successively in the resulting equation, we obtain two equations, from 

which 7y and oy can be found : the hypergeometric functions reduce to the 

type 

F(u, V, w, 1), 

which in § 135 was shewn to be expressible in terms of Gamma-functions, and 
the type F {Uy v, w, 0), which clearly has the value unity. 

As already explained, in certain cases (e.g. when one of the exponent-differences is an 
integer) the above theory of the solutions requires modification. For a discussion of these 
cases the student is referred to Lindel6f s paper already mentioned, and Klein's Lectures 
" Ueber die hypergeometrische Function." 

142. Solution of the general hypergeometric differential equation by 
a definite integral. 

We next proceed to establish a result of great importance, relating to 
the expression of the hypergeometric function by means of definite integrals. 

Let the dependent variable y in the differential equation of the general 
hypergeometric function ((A) of § 137) be replaced by a new dependent 
variable /, defined by the relation 

y^iz-aYiz-hf (z-c)y I. 

The differential equation satisfied by / is easily found to be 



1 4-/9-/8' . 1 + 7-7] 



d*/ fl+a-a l4-i8-/8' 
dz^ ( z^a z — o 



z — o 



dJ 
dz 



(a + /3 + 7){(tt4-/3 + 7 + l)^ + Sa(a + /y-fy-l)} 
"^ {z-a){Z''b){Z'-c) 



which can be written in the form 



i 



(LJMi_l) p. (^) + (f _ 1) ^> (^) 



/. 



where { f = 1 -a-/3-7 = o' + /3' + 7', 

Q{z) = {z-a)iz-h){z-c), 
R(z) = 2 (a + y3 + 7) (z -b)(z- c). 

It must be observed that the function / is not regular at e.'oo , and consequent! j the 
above differential equation in / is not a case of the generalised hypergeometric equation. 

We shall now shew that this differential equation can be satisfied by an 
integral of the form 



254 TRANSCENDENTAL FUNCTIONS. [CHAP. XI. 

provided the path C of integration is suitably chosen. 

For on substituting this value of / in the differential equation, the 
condition that the equation should be satisfied becomes 

= f (^ - ay+P+y' (t - 6)*+^'+r-i (t - cy-^P+y'-i {z - ^)-«-^-y-2 xdt, 
J c 

where 

+ {t-z)[R{z) + {t-z)R{z)] 
= {i-^)[Q(t)-{t-zy\ + it-z){R(t)-{t-zr^(<^ + p + i)] 
^{^-^)Q{t) + {t-z)R{t) 

= -(l + a + /3 + 7)(<-a)(<-6)(<-c) 

+ 2(o' + /3 + 7)(<-6)(«-c)(«-«), 
or Z = (< - a)'-"'-*-y (« - hy-^-^-y (t - cf-^-f-y' (z - <)«-H>+y+« 

^ {(« - aY'+^+y (t - by+^'+y (t - c)'+p*y' (t - ^)-o+«+p-h')). 

It follows that the condition to be satisfied reduces to 

dV 



o-lis"-!/^- 



where F= {t - a)*'+^+y {t - 6)«+^'+> (< - c)*+^+>' (^ - 5)-ti+«+^+y). 

The integral / will therefore be a solution of the differential equation, 
provided the path of integration C is such that the quantity V resumes its 
initial value after describing the arc C. 

Now F= (t - ay-^^-^r-^ (t - 5)*+^'+>-i {t - c)»+^+>'-^ (z - 0"^"^"^ ^, 

where U^it-a) (t- b) (t - c) (^ - 1)'^ ; 

and the quantity U resumes its original value after describing any contour : 
hence if (7 be a closed contour, it must be such that the integrand in the 
integral / resumes its original value after describing the contour. 

Hence finally any integral of the type 

{z^aY{z^hy{z''C)y f (t-ay-^-^^'-'(t-b)y-^'^P''Ht-cY-^-^'-'(z-t)'^-fi-ydt, 

J c 



142] HTPERGEOMETRIC FUNCTIONS. 255 

where C is either a closed contour in the t-plane such that the integrand 
resumes its initial value after describing it, or else is an arc such that the 
quantity V has the same value at its termini, is a solution of the differential 
equation of the general hypergeonietric function. 

Example 1. As an example, we shall now deduce a real definite integral which (for a 
certain range of values of the quantities involved) represents the hypergeometric series. 

The hypergeometric series F(a, by c, z) is, as already shewn, a solution of the differential 
equation of the function 

' 00 1 

P ■ a t 

1-c b c^a-b 
The int^ral 

thus becomes in this case 

I <«-«(<-l)«-*-i («-«)-« eft. 

Now the quantity F is in this case 

and this tends to zero at ^b 1 and «= oo , provided c> b>0. 

Hence if these conditions are fulfilled, we can take as the contour C an arc in the 
^-plane joining the points t^l and <nao ; so that a solution of the differential equation is 



/ 



00 

f^ if- l)«-*-i {t-z)-^dt. 
1 



In this integral, write t^-; the integral becomes 



/ 







this integral is therefore a eolutum of the differential eqtiatumfor the hypergeometric series. 

It is easily seen that this integral is in fact a mere multiple of the hypergeometric 
series 

jP(a, 6, c, z) ; 

for supposing | « | < 1, and expanding the quantity (1 - uz)'^ in ascending powers of z by 
the Binomial Theorem, the integral takes the form 

fV»(l-«)'-^»du+ i «(«+l)-(«+*-l) ^[V-'*r(i_^)«-t-.rf,^ 
Jo r=l T' Jo 



or 



r«i ri 



256 



TRANSCENDENTAL FUNCTIONS. 



[chap. XL 



or 
or 



nth . m/io. ; a{a'\-l)''>{a+r-\)b{h+\).,.{h'\-T-\) \ 



a(a+l)...(a+r-l)6(6+l)...(fe-hr~l) 
r!c(c+l)...(c+r-l) 

5(6, c-6)jP(a,6, c,z), 
which establishes the result stated. 



Example 2. Deduce SchUiflUs integral for the Legendre functions^ as a case of the 
general hypergeometric integral. 

Since the L^;eQdre equation corresponds to the hypergeometric function 

-1 00 1 ^ 

n+1 £ V, 

. -n J 
the corresponding integral is 



or 



[ (^-l)«(«-0)-'»-icfe, 



taken round a contour C such that the integrand resumes its initial value after describing 
it ; and this is Schlafli's int^ral. 

Example 3. Deduce Laplace's integral for the Legendre functions, as a case of the 
general hypergeometric integral. 

If we write 

«=i(«*+r*), 

4 

the Legendre differential equation becomes 



^-i-P I ^ \^y n(ti+l) y 



This corresponds to the hTpergeometric function 

.0 00 1 



n 
"2 


n + 1 
2 


f 


- y 


n+1 
2 


n 
2 




1 





and so the hypergeometric integral becomes in this case 

("i I tt» ( 1 - u) "* (i - w)"* du, 
taken round a contour enclosing the points u=l and u^(. 



Write 



Then the integral becomes 



taken round a contoiu* enclosing the points u^l and u^CK 






143] HYPERGEOMETRIC FUNCTIONS. 257 

Write u=sA( in this integral ; we thus obtain 

(l-.2«A+A«)-U»cM, 



/' 



the integral being now taken round a contour in the A-plane enclosing the points k=( and 

Suppose now that the real part of z is positive ; and let the contour become so attenuated 
as to reduce to a small circle surrounding the point h=(^ another small circle surrounding 
the point h=(~\ and the line joining the points ( and f "^ described twice. The small 
circles contribute only infinitesimally to the integral, which thus becomes a multiple of 



/ 



^^(1-2M+A«)-U«(i^ 



Writing A =«+(«•- 1)* cos 

in this integral, we obtain 



/; 



{«+'(««-l)*C08<^}*rf<^, 





which is one of Laplace's integrals (§ 119). 

143. Determination of the integral which represents P<*^ 

We shall now shew how the integral which represents the particular 
solution P<*^ (§ 141) of the hypergeometric diflferential equation can be 
found. 

We have seen (§ 142) that the integral 

7=(^-a)«(^-5)^(^-c)vf (^-a)^+y+«'-H«-6)y+*+^'-H«-c)*'^^'-K'^-0"*"'^"^* 

J c 

satisfies the diflferential equation of the hypergeometric function, provided 
(7 is a closed contour such that the integrand resumes its initial value after 
describing C, Now the singularities of this integrand in the ^plane are the 
points a, b, c, z; and on describing a simple closed contour enclosing the 
singularity 6 alone, the integrand resumes its initial value multiplied by 

I Jiniy+a+fi'-l) 

as is seen by writing it in the form 

(^+y+tt'-l)log«-a)+(y+a+/3'-l)log(f-6)+(«+/8+y-l)log(<-c)-(a4-/3+Y)log(«-0 

e . 

Take then a point in the ^-plane, and draw a loop in the ^-plane passing 
through and encircling the point 5, but not encircling any of the points 
a, c, z. Let an integral taken in the positive or counter-clockwise direction 
of circulation round the perimeter of this loop be denoted by the sign 

Jo • 
W. A, 17 



258 TRANSCENDENTAL FUNCTIONS. [CHAP. XI. 

and let an integral taken in the negative direction of circulation round the 
perimeter of the loop be denoted by 

r(6-) 
so that we have the equation 



/•(&+) r(b-) 

Jo Jo 



where it is understood that the initial value of the integrand in the second 
integral is taken equal to the final value of the integrand in the first 
integral 

Let now a contour C be drawn in the following way. Take first a loop 
starting from 0, encircling the point 5 in the positive direction, and returning 
to ; then a loop starting from 0, encircling the point c in the positive 
direction, and returning to 0; then a loop encircling the point 5 in the 
negative direction ; and lastly a loop encircling the point c in the negative 
or clockwise direction. 

Conformably to the notation already explained, an integral taken round 
this contour will be denoted by 

{b+, c+, 6-, C-) 



Jq 



Now after description of this contour, the integrand of the integral / 
already considered resumes its initial value multiplied by 

gaIrf(y+tt+/8'-l-hE+^47'-l-y-«-^'+l-«-^-y'+l) 

or 1, i.e. the integrand resumes its initial value*. 

Hence if C be taken as the contour, the integral / will satisfy the 
differential equation. 

Thus 

/•(6+,c+.6-.c-) 

J = (^ - aY (z -hfiz" c)y (t - ay+y +«'-' (t - 5)y+-+^'-^ 

Jo 

satisfies the differential equation of the hypergeometric function. 

Now suppose that the point z is taken near to the point a, so that |z — a| 
is less than either |6 — a| or |c — a|. We can clearly draw the contour just 

* These double-circuit inUgraU were introdaced by Jordan in 1887. Clearly any namber of 
contours can be formed in this way, it being necessary only to ensare that each singular point is 
encircled as often in the negative or clockwise direction of circulation as in the positive or counter- 
clockwise direction. 



144] HTPEBQEOMBTRIC FUNCTIONS. 259 

described in such a way that, for all points ^ on it, I ^ — a | is greater than | ^ — a | . 
Thus we can write 



Jo 



(. . ^-, (, - izf) 



z - a\-*-^-y 



dt 



Under the conditions already stated, each of the expressions 

can be expanded by the Binomial Theorem in ascending powers of (z — a). 
We thus obtain for I an expansion of the form 

/ = (2r - a)« {4 + £ (ar - a) + (7 (^ - a)« + . . . }, 

and as / satisfies the differential equation it must therefore be a multiple of 
the particular solution P**^ of § 141. 

Thus 

pw = Constant x (^ - a)« (z -hfi^z- c)^ {t - af-^y^'"^ 

Jo 

(t - 6)y+«+^ -> (t - c)-^+y'-i (z - 1)-^-^-^ dt 
Similarly 

r(6+.c+,6-,c-) 

P^*') = Constant x (z - a)*' (z ^hf{z- c)y {t - ay+y+— ^ 

In the same way the particular solutions P^\ P<^'»,*Pw, P<y'>, can be 
expressed as contour-integrals. 

144. Evcduation of a dovhU-contour integral. 
We may note that an integral 

/; 

can be expressed in terms of the integrals 

in the following way. 

Let the initial value of the integrand at the point be denoted by T. After describing 
the loop roujid a, the integrand will have at the value g^'^^^+^+V"^) Ty and the part 

17—2 



■(o-»-,6+, o-,6-) 



260 TRANSCENDENTAL FUNCTIONS. ' [CHAP. XI. 

I of the integral I will have been obtained. Describing next the loop 

/(o+, 6 + . 0-, 5-) 
will therefore be 



^»i(a+/r+y- 



■'/r*'' 



and the integrand will return to with the value 

^ (tt'+/8+y-l+e+^'+y-l) qt 

Describing next the loop round a in the negative direction, we observe that the corre- 
sponding part of the integral would have been 



r: 



if the integrand had had for initial value 

which is its final value when the loop is described with the initial value ^ : it is therefore 
actually 



or — e 



jo ' 

i. ' 



_ 21^i(«+^'+y-l) 



and lastly, describing the loop round h in the negative direction, we obtain the part 

of the integral. 

Collecting these r^ults, we have 

/■''•+'*+'»-'*-L(i-«««t«+c'+y-i>) /■'"*'_(! _««'rt(«+o'+y-i)) /■•***, 

a formula which furnishes the value of the double-contour integral in terms of two simple- 
contour integrals. 

146. Relations between contiguous hypergeovietric functions. 

Let P (z) be a hypergeometric function with the argument z, the singu- 
larities a, 6, c, and the exponents a, a',/8, I3',y,y. Let Pi+i^rn-i(z) denote the 
function which is obtained by replacing two of the exponents, I and m, in 
P(z) hy I + 1 and m —1 respectively. Such functions Pi+i^fn^i{z) are said 
to be contiguous to P (z). There are clearly 6 x 5 or 30 contiguous functions, 
since I and m may be any two of the six exponents. 

It was first shewn by Riemann* that the function P(z) and any two of 
its contiguous functions are connected by a linear relation, the coefficients in 

which are polynomials in z, 

» 

• Abhandlungen der Kdn, Qet. der Wiss. zu O&ttingen, 1867. 



145] HTPERGEOMETRIC FUNCTIONS. 261 

80 29 
There will clearly be — ^ — or 435 of these i*elations. In order to obtain 

them, we shall take P (z) in the form 

P (z) = (^ - a)* (^ - by (z - c)y lit" df-^-^'-^ (t - 5)y+«+^'-i 

Jc 

(t - cy+^+y'-i (t - ^)— ^ dt, 

where C may be any closed contour in the ^-plane such that the integrand 
resumes its initial value after describing C, 

First, since the integral round C of the differential of any function which 
resumes its initial value after describing G is zero, we have 

= J ^ {(^ - ay-^f"^ (t - 6)-+/»'+r-i (t - c)«+^+y-» (t - r)-H»-y} dt, 

or 

= (a + ^ + 7) f (« - ay-^+r-' (t - ftV+^'+y-i (^ - cy-^P+y'-i (t - -?)-- ^-r dt 

Jo 

+ (a + )8' + 7 - 1) f (« - a/'+^+T' (e - 6)*+/»'+y-« (« - c)*+^+r'-i (t - «)-«H»-y (ft 
+ (a + )8 + 7' - 1) f (t - a)*'+^+y ft - 6)•+^'+r-l rt - cV+^+y-i (^ - ^)-Hi-y d^ 

- (a + )8 + 7) f (« - a)*'+^-h' (e - 6)*+^'+y-» (t - c)-^^+r -^ (« - ^)-Hi-y-i d«, 

or 

(a' + )8 + 7)P + (a + )8' + 7-l)P.'+i.r-., + (a + )8 + 7'-l)P.'+,.y-i 



(a-h^ + 7) p 



^-6 



/l+l.Y'-l. 



Considerations of symmetry shew that the right-hand side of this 
equation can be replaced by 






jer — c 



These, together with the analogous formulae obtained by cyclical inter- 
change of (a, a, a) with (b, /8, )8') and (c, 7, 7O, are six linear relations 
connecting the hypergeometric function P with the twelve contiguous 
functions 

P«+l.A'-l» P/M-1,y'-1> Py+1,«'-1> Ptt+l,y-l> Pi8+1,«'-1, Py+l.A'-l* 

Pa'+i.ir-i, P«'+i.y-i, P/i'+i^y-i, P/i'+i,.'-!, Py+i,.'-!, Py+i,/i'-i. 



262 TRANSCENDENTAL FUNCTIONS. [CHAP. XI. 

Next, writing < — a = (^ — 6) + (6 — a), and using Pa'-i to denote the result 
of writing a' — 1 for a' in P, we have 

Similarly P = Pa'_i,y+i + (c - a) P.-i. 

Eliminating P.'-i from these equations, we have 

(c - 6) P + (a - c) P«'-i. fi^+, + (6 - a) Pa'-i.y+i = 0. 

This and the analogous formulae are three more linear relations con- 
necting P with the last six of the twelve contiguous functions written above. 

Next, writing (^ — jer) = (^ — a) — (jer — a) we readily find the relation 
P = ^j P/1+i.y.i - (^ - aY^' (z - b)f^ (z - c)y 

xf(t- ay-^-^'-^ (z - a)y+«+/»'-^ (z - 6)«+^+y-i (^ - ^)-«H»-y-i d^, 

which gives the equations 

{z - a)-» {P - (^ - 6)-^ J'/i-M.y-il = (^ - 6)-^ [P-i^' or' Pr+i..'-il 

- (^ - c)-^ {P - (^ - a)-> P.-M.^'-i}. 

These are two more linear equations between P and the above twelve 
contiguous functions. 

We have therefore now altogether found eleven linear relations between 
P and these twelve functions, the coefficients in these relations being rational 
functions of z. Hence each of these functions can be expressed linearly in 
terms of P and some selected one of them ; that is, between P and any two of 
the above fwnctions there exists a linear relation. The coefficients in this 
relation will be rational functions of z, and therefore will become polynomials 
in z when the relation is multiplied throughout by the least common multiple 
of their denominators. 

The theorem is therefore proved, so far as the above twelve contiguous 
functions are concerned. It can in the same way be extended so as to be 
established for the rest of the thirty contiguous functions. 

Corollary, If functions be derived from P by replacing the exponents 
a, a', )8, )8', 7, </, by a+p, «' + ?, )8 + r, /S^H-*, 7 + ^, 7 +ti,where p, 5, r, «, f, w, 
are integers satisfying the relation 

then between P and these functions there exists a linear relation, the co- 
efficients in which are polynomials in z. 



MISC. BXS.] HYPERGEOMETRIC FX7NCTI0NS. 263 

This result can be obtcdned by connecting P with the two functions by 
a chain of intermediate contiguoi^s functions, writing down the linear 
relations which connect them with P and the two functions, and from these 
relations eliminating the intermediate contiguous functions. 

It will be noticed that many of the theorems found elsewhere in this 
book, e.g. the recurrence-formulae for the Legendre functions (§. 117), are 
really cases of the theorem of this article. 

MlSOELLANEOUS EXAMPLES. 

1. Shew that 

c 

2. Shew that 

jP(a+l, 6+1, c, t)-F(a, 6, c, t)^^F{a+l, 6+1, c+l, a). 

3. If P(«) be a hypergeometrio function, express its derivates -^ and -^ linearly in 

terms of P and contiguous functions, and hence find the linear relation between P, -^ , 

cPP 
and -rj , i.e. verify that P satisfies the hypergeometric dififerential equation. 

4. If 

W(a,b,x) denote ^(-^i h 2, -bx\ 

shew that the equation y= TF(a, 6, a?) 

is equivalent to a?-« Tr(6, a, y). 

5. Shew that a second solution of the differential equation for 

F (a, 6, c, s) 
is a;»-«F(a-<J+l, 6-c+l, 2-c, x). 

6. Shew that the equation 

(«2+V)^+(«i-»-V)^+K-»-Wy=o 

can, by change of variables, be brought to the form 

and that this latter equation can be derived from the hypergeometrio equation 

by the substitution 6^^, x— , where m is infinitely large. 



264 



TRANSCENDENTAL FUNCTIONa. 



[chap. XI. 



7. Shew that 



o:(z)^p 



/ -1 00 1 



I 



— « 



where C («) is the coefficient of A*» in the expansion of (1 - 2A«+A*)-«' in ascending powers 
of A. 

8. Shew that, for values of x between and 1, the solution of the equation 

where il, ^, are arbitrary constants and F{af /3, y, x) represents the hypergeometric series. 

(Cambridge Mathematical Tripos, Part I, 1896.) 

9. Shew that the differential equation for the associated Legendre function Pi!!*^{z) 
of order n and degree m is satisfied by the three functions 



( 



00 



\ 



1 
2 



p \ « W* ""^ — w 



1 1-2 
2^ ^" 



v-s"^ n+1 -^m 



00 



P\ 



n 
~2 

n-hl 
V 2 



2 «-(««-l)* V, 



— m 



n+1 



/ 00 1 



P\ 



n m - 1 
2 2 " l3^ 



n+1 9n 1 
V"T" "2 2 



10. Shew that the hypergeometric equation 



(Olbricht) 



^(^-i)S-{y-(«-»-i3+i)^}2+flft^-o 



is satisfied by the two integrals 

/ f^-Vl-«)T^-^-^(l-a?«)-*(i« 

J 



and 



//-*<- 



f)— >{l-(l-«)f}— A. 



MISC. EXS.] HYPBRGEOMETRIC FUNCTIONS. 265 

11. If 

(l-ar)*-^-T'i?'(2a, 2/3, 2y, ^) = l+^ar+Cx«-|-2>a?8-|-...,• 
8hew that 

F{a, ft y+i, a?)/'(y-a, y-ft y+i, x) 

''^+y+i^^y+i)(y+t)^^ ^ (y+i)(y+i)(y+|)^'^^ •- 

(Cayley.) 

12. Prove that 

P^ («)=! tannir {§, «- Q.^.i («)}, 

where Pm(2) and Qn(^) cure the Legendre functions of the first and second kind of 
order n. 

13. If a function F{a^ /S, /S', y ; x^ y) be defined by the equation 

F{a, ft /y, y ; x, y)° r(o)rty-„) /,*«""' (1 -«)'^~' (1 - «*)"* (I -^yT^du, 

then shew that between /* and any three of its eight contiguous functions 

F{a±l\ F{fi±\\ F{^±1\ F(y±l), 

there exists a homogeneous linear equation, whose coefficients are polynomials in x and y. 

(Leyayasseur.) 

14. If y — a - 3 < 0, shew that, for values of x nearly equal to unity, 

and that if y— a— ^aO, the corresponding approximate formula is 

JPf a^ ^\ r(a+g) , 1 
^(o,fty,^)«- r(a)r03) ^^i^^- 

(Cambridge Mathematical Tripos, Part II, 1893.) 

15. Shew that when \x\ < 1, 



/; 



^^-'(v-ar)'*-*-!^— ^(l-v)-*'Ji' 



«--46'^sinair8in(p-a)ir. ^^^^°^,^^°^ i?^(a,ai,p,^X 



where c denotes a point on the finite line joining the points 0, x, the initial arguments of 
y-^ and of i^ are the same as that of ^, and that of (1 - v) reduces to zero at the origin. 

(Pochhammer.) 



CHAPTER XII. 
Bessel Functions. 

146. Ths Bessel coefficients. 

In this chapter we shall consider a class of functions known as Bessel 
functions, which present many analogies with the Legendre functions con- 
sidered in Chapter X. As in the case of the Legendre functions, we shall 
first introduce the functions, or rather a certain set of them, as coefficients 
in an expansion. 

For all finite values of z, and all finite values of t except t » 0, the 
function 

can be expanded by Laurent's theorem (§ 43) in a series of ascending and 
descending powers of t If the coefficient of <^, where n is any positive 
or negative integer, be denoted by Jn {z), we have (by § 43) 



Jn{z)^^\yr---e^'^'"'^du, 



the integral being taken round any simple contour in the u-plane enclosing 
the point t^ = 0. 

To express this quantity Jn (z) as a power-series in z, write 

2t 



z 



Thus '^'^(')-L{i)'h~^''*'^'^'' 

the integral being taken round any simple contour in the ^plane enclosing 
the point t^O. This can be written 

-.(')-^(l)M.^'®7'— '^ 



146] BESSEL FUNCTIONS. 267 

Now (§ 56) we have 
^— . I ^"^'^Vd^sa the residue of the function ^-»--*^» e* at its pole, the origin. 

If n is a positive integer, this residue is 

1 

if n is a negative integer, say = — «, the residue is zero when r = 0, 1, 2, . . . « — 1, 

and when r ^ « it is 

1 

In any case, the residue is 



Thus if n is a positive integer, we have 






and if n is a negative integer, equal to -- 8, we have 

or Jn(z)=^('-iyJ.(z). 

Whether n be a positive or negative integer, the expansion can clearly 
be written in the form 

~ ^^ rto 2'^*-r I r (n + r + 1) • 

The function Jn{z) thus defined for integral values of n is called the 
Bessel coefficient of the nth order. 

We shall see subsequently (§ 149) that the Bessel coefficients are a particular case of a 
more extended class of functions known as Bessd fuTtctions, 

Bessel coefficients were introduced by Bessei in 1824 in his " Untersuchung des Theils 
der planetarischen.Stdrungen, welcher aus der Bewegung der Sonne entsteht." 

In reading some of the earlier papers on the subject, it is to be remembered that the 
notation has changed, what was formerly denoted by Jn {z) being now denoted by J^ (2«). 

EaiompU 1. Prove that if 

(l-2a^-^)H46«^ ^■*" "^^ "^^ ••• ' 

then will ««8in6«=^i/i(«)+^2/,(«)+^s/j(«)+.... 

(Cambridge Mathematical Tripos, Part I, 1896.) 



268 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 

For replacing the Bessel functions in the given series by their values as definite 
integrals, we have 






^(tR 



2iriy / 2a 1 Y 46» ^ 



the integrals being taken round any simple contour in the u-plane enclosing the origin. 
Taking a new variable t, defined by the equation 

vre thus have 



^.7i(z)+il^,« + ^j/,W + ...-2^.j" 



<«+6« ' 



where the int^;ration is now to be taken in the clockwise direction round any large simple 
contour in the ^plane. This expression is (§ 56) equal to minus the sum of the residues 
of the function 

at its poles t^ib and t=-ih ; that is, it is equal to 

2t 2i 

or 0" sin bzj 

which is the required result 

Example 2. Shew that, when n is an integer, 

mss— 00 

We have J^^^l^'D ^^i''^ Ji^^ , 

or 2 <"/,(* +y)= 2 <"/«(«) s rJr(y). 

»— — 00 ms-oo ra— 00 

Equating coefEicients of t^ on both sides of this equation, we have the required 
result. 

147. BeaseVa differential equation. 

We have seen that, for all integer values of n, the Bessel coefficient 
of order n is expressed by the formula 

where C is a simple contour in the ^-plane enclosing the point ^ « 0. 



147, 148] 



BESSEL FUNCTIONS. 



269 



We shall now shew that the function Jn{z) is a solution of a certain 
linear differential equation of the second order, namely, 



For we find in performing the differentiations that 

-7r-7ik \ t-^''^e «1- 

27rt2~Jc I 



n4-l ^ 
t "^4^ 



dt 



= 0, 



t-^. 



since the function e ** f^"^ resumes its original value after the point t has 
described the contour in question. 

Thus Jn {z) satisfies the differential equation 



d^Jniz)^ldJn(z) 



dz' 



dz 



+ ^i-gj„(.) = o. • 



This is called BesseVs equation of order n. Its properties in many respects 
resemble those of Legendre's differential equation, which is also a linear 
differential equation of the second order. 

148. BesseVs equation as a case of the hyper geometric equation. 

If c be any finite quantity, the differential equation of the hypergeometric 
function 

/ 



n 



00 



tc T^-^'^c z 



) 



— 71 — IC 2 ^ *^ 



) 



is (§ 137) 



dz^ z dz \^z z — c J 



z(z — c) 



If in this equation we make c tend to an infinitely large value, we obtain 



270 



TRANSCENDENTAL FUNCTIONS. 



[chap. XII. 



which is Bessel's equation of order n. Thus BeswVs eqwUion ca/n be regarded 

as a limiting case of the hypergeometric equation, corresponding to the 

function 

/ X c \ 



Limit P { 



n 



1 . 
ic 2 "^ *^ ^ 



0sac 



\ 



— n — ic 2 "■ *^ 



Another representation of BesseVs equation as a limiting case of the 
hypergeometric equation is the following. 

If we change the dependent variable in Bessel's equation, by writing 
e^u, the differential equation for u is easily found to be 



y= 



dz 



I /^. 1\ du fi n*\ ^ 



Now if c be any quantity, the differential equation of the hypergeometric 
function 



P\ 







n 



00 

1 
2 







8 



— n I — 2tc 2ic — 1 



IS 



dhi 
d^ 



^ /I I 2 - 2ic\ du ^ /n^c ^ 3 ^\ u ^^ 
\z z — c J dz \ z 8 ) ziz — c) 



If in this equation we make c tend to infinity, we obtain 



cPu /I . o .\ du [ n« i\ ^ 



which is the above equation. Hence Bessel's equation is a limiting case 
of the hypergeometric equation, being the equation for the function 



e^ Limit P . 







n 



oc 

1 

2 







8 



— w T — ^io 2tc — 1 

4 



Bessel's equation is connected not merely with the general hypergeometric 
equation, but with that special form of it which we have considered in con- 
nexion with the Legendre functions. 



148] 



BESSEL FUNCTIONS. 



271 



For the differential equation of the associated L^eudre function (§ 129) 
is (§ 138) the equation of the function 



P^ 



-1 



X 



m - 

2 " + 1 



m 
I 2 






m 
2 

m 



1- 



_ 
2n« 



or (§ 139) 



/ 4n« 



00 







< 



\ 



m 
2 

m 
1 



n + 1 ^ -^ 



m 
2 



— n — 



m 
2 



The differential equation of this function is 

*y / 1 1. \ c?y / m» __ n + 1 m|\ nh/ 

i{z^yW -4ffi^ zV d{z^)\ z^-4m,^ n z^J 2^{z^'-An^) 

If in this equation we make n tend to infinity, it becomes 



=0. 



d{z^y^ 2^d{z 



^l dy 



or 



r)-(->-?)|.=». 



which is Bessels equation. Thus Bessels equation of order m is the same 
as the equation for the function 

Limit PrT (l - 



2nV' 



By considering Bessel's equation as a limiting case of the hypergeometric equation, 
we can deduce certain solutions in the form of definite integrals. 

For the differential equation of the function 

'^ 00 e 






ic 



is satisfied by the integral 

'(-:-y7.'-'(-0 



^\»+i-w« 



{t-zy^-hdiy 



272 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 

if C is a ooDtour such that after describing C the integrand returns to its initial value. 
When c becomes infinite, this expression reduces to 



sf^tr*^ j <-»-i(<-«)-»-ie«'cfc, 



which accordingly satisfies BesseFs equation if C be a contour of the kind described ; C can 
for instance be a figure-of-eight contour encircling the points t=^0 and t=:z. 

In fact, if we write 



we have 






=0. 
Other solutions can be found by changing the signs of n and t. 

Example. Shew that Bessel's differential equation is the limiting case of the equation 
of the hypergeometric function 

00 c* 

P\ in i(c-n) ^ 

when c tends to infinity. 

149. The general solution of BesseVs equation by Bessel functions whose 
order is not necessaHly an integer. 

We now proceed, in the same way as in § 116, to extend our definition of 
the function «/n(^) to the general case in which n is not an integer. 

It appears from the proof given in § 147 that, whatever n may be, the 
differential equation 

is satisfied by an integral of the form 

y=z'' r'*-^ e *^ dt, 

J c 

provided the path of integration C is a contour on the ^-plane, so chosen that 
the function 

e «r^-i 
resumes its initial value after describing C. 



149] BESSEL FUNCTIONS. 278 

Now when the real part of Ms a very large negative number, the 
function 

is infinitesimal. Hence y will be a solution of the differential equation, 
provided the contour G begins and ends with values of t whose real part 
is infinitely large and negative. 

Let therefore a contour C be taken which begins at the negative end 
of the real axis, and after proceeding close to the real axis to the neighbour- 
hood of the origin makes a circuit of the origin and returns, close to the real 
axis, to the negative end of the real axis again. The integral y taken round 
this contour satisfies BesseVs differential equation. 

We shall now shew that this solution y can be expressed in the form of 
a series of powers of z. 

Suppose as usual that by tr^^^ is understood that branch of the function 
ir^-^ which when continued (§ 41) to the point ^ = 1 by a straight path, 
arrives at the point ^ = 1 with the value unity. 



Then we have 



y = £«| r^>e^6 **<fo 



00 / ly-.sr-fn r 

r=o ^.r\ Jc 
But (§ 100) we have 

But when n is an integer, we have (§ 146) 

^"^^^^ ^to2^^r! r(n + r + l)* 
Comparing these results, we have, when n is an integer, 



•^" <^> = 2^- (l)7c'"^'*'"^'"' 



where C is the contour already described. 

Now we have seen that the right-hand side of this equation has a meaning 
and satisfies BesseFs differential equation for all values of z and all values of n ; 
whereas, up to the present, Jn{z) has been defined only for integral values 

w. A. 18 



274 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 

of n. We shall take this opportunity of extending the definition of Jn (z), in 
the following way. 

For aU valuea of n and of z, the function 

h 'Silr''"'"' 

CO (,l)r^gir+n 

rro2~+*-r! r(w + r-!-l) 



or 



will he denoted by Jn (z). In the integral, tr^"^ is to have the value which 
becomes unity when the variable t travels in a straight line to the point ^ = 1 : 
and is a contour which encircles the point ^ = and begins and ends at 
the negative end of the real axis in the ^plane. The function Jn(z) thus 
defined is called the Bessel function of z, of the first kind and of the order n ; 
it satisfies BesseVs differential equation of order n. 

Since Bessel's differential equation is unaltered by the change of n into 
— n, we see that J^n(z) is also a solution of the equation ; and therefore the 
general solution of BesseVs equation is of the form 

aJn (z) + bJ^n (z\ 

where a and b are arbitrary constants, except in the case in which Jn (z) and 
«/Ln (z) are not independent functions ; this exceptional case happens when n 
is an integer, for then, as we have already seen, we have the relation 

Jn (Z) = (- irJ-n (*). 

A second solution of BesseFs equation in the case when n is an integer 
will be given later. 

160. The recurrence-formulae for the Bessel functions. 

As the Bessel functions, like the Legendre functions, are members of 
the general class of hypergeometric functions, it is to be expected that 
recurrence-formulae will exist between them, corresponding to the relations 
between contiguous hypergeometric functions (§ 145). 

We shall now establish these recurrence-relations ; the proof given does 
not assume the order n to be an integer, and consequently the formulae are 
valid for all values of n, real or complex. 

Let C be the contour described in the last article, which begins and ends 
at the negative end of the real axis in the ^plane, and encircles the point 
t = 0. 



Then since the function 



e ^ir^ 



¥mmt^mBsm 



150, 151] BESSBL FUNCTIONS. 275 

is infinitesimal at the extremities of this contour, we have the equation 

2n 

or */n-i (^) + */n+i (-8^) = — t/n (-2) (A), 

z 
which is the first of the recurrence-formulae. 

Next we have, by differentiation, 



or 






''-^.- -,■''<.')- J-^M (B). 

From (A) and (B) it is easy to derive other recurrence-formulae, e.g. 

^-i^ = \[j^,{z)-j^^,{z)] (C), 

and ^!^) = j^,(,)_^/,(,) (D). 

Example 1. Shew that 

16'^^=^.-4(«)-4^«-,W+6/»(«)-4y,^,(*)+J,^,W. 
Example 2. Shew that 

161. Relation between two Bessel fimctions whose orders differ by an 
integer. 

The various recurrence-formulae found in the last article can* however be 
easily deduced from a single equation, which connects any two Bessel 
functions whose orders differ by an integer, namely 



n+r 



w^/ ly _*; 



(-ly 






z"^^ "" ^ (zdzy 

where n is any number (real or complex) and r is any positive integer. 

18—2 



276 



TRANSCEKDENTAL FUNCTIONS. 



To establish this result, we have, by § 149» 



dr {Jn{z) 



d{s^y\ z"" 



dr I f 

d{j^y27n.2''Jc 



e *tdt 



dt 









[chap. XII. 



(- 2/ -2^+'" 



•^ n+r V-*^)* 



which is the equation required. 



The recurrence- formulae can be derived without difficulty from this resultw 
Thus, equation (B) of the last article is obtained by taking r = 1 in this 
equation : and equation (A) of the last article may be derived in the following 
way. 

Taking r= 1 and r = 2 successively in the formula just proved, we can 
express the first and second derivates of Jn(z) in lerms o{ Jn{z), Jn+i{z) and 
Jn+2(z), in the form 

dJn (z) _ n J. ,. J , . 

— ^^ =- - (1 - n)Jn (Z) y-^n+i W + «/n+s(4 

Substituting these values in BesseVs equation 



we have 






Changing n to (n — 1) in this result, we have 

J,+,(z)-^/,(*)+J._,(^) = 0, 

z 
which is the formula (A) of the last ai*ticle. 

The other recurrence-formulae can be derived in a similar way. 



152, 153] BESSEL rUNCTIONS. 277 

162. The roots of Besael fwncHons, 

The relation established in the preceding article enables us to deduce the 
interesting theorem that between any two consecutive recU roots of Jn {z) there 
lies one and only one root of Jn+i (^)*. 

For since Jn(z) satisfies BesseFs equation, it follows that the function 
y = Jir^ Jn (z) satisfies the differential equation 

From this equation it is evident that if f be a value of z (real and not 
zero) for which ^ is zero, then the signs of -~ and y must be unlike at the 
point 2r = f . Now let -^ = ^i and ^ = f, be two consecutive roots of the 
function -^. It is clear from the differential equation that neither y nor -~ 

can be zero at either of these points. Then the function -^ -~ has a 

different sign just before reaching 2^ = fa to that which it has just after 

leaving -? = fi; and hence it follows that the function y -^ has a different 

sign just before reaching z= ^^to that which it has just after leaving z^ fj. 
The function y must therefore have an odd number of roots between the 
points 2r = f 1 and z = fj. 

But from Rollers Theorem it follows that y cannot be zero more than 
once in this interval : so y must have one and only one zero between the 

points 2^ = f 1 and 2^ = fa • and therefore the zeros of y and of -p occur 
alternately. 

Thus, between any two consecutive roots of the function z~^ Jn (z) there 

lies one and only one root of the function -j- {z~^ Jn (z)] or - z~^ J^^i (z): which 
establishes the theorem. 

163. Expression of the Bessel coefficients as trigonometric integrals. 

We shall next obtain a form for the Bessel coefficients (ie. the Bessel 
functions for which the order n is an integer), which in some respects 
corresponds to the Laplacian integrals obtained in ^ 119 and 132 for the 
Legendre functions. 

* The proof here given is dae to Oegenbaaer, MonaUhefU fiir Math, ym. (1897). 



278 TRANSCENDENTAL FDNCTIONS. [CHAP, XII. 

If Id the equation 
we write t = e^, we have 

n»-ao 



Changing i to — i in this equation, we have 



»« -00 



Adding and subtracting these results, we have 

00 

COS {z sin ^) = 2 J^ (z) cos 7uf>, 



n« - 



sin (z sin <^) = S Jn (z) sin n^. 



n«-flo 



Since J^ (z) = (— l)'*«7Ln (A these equations give 

cos(* sin <l>)^Jo (z) + 2J, (^r) cos 2^ + 2J4 (2r)c08 4<^ + ..., 
sin {z sin ^) = 2Ji (^) sin ^ + 2/, (5) sin 3^ + ... . 

As these are Fourier series, we have (§ 82) 

J^ (-?) = - I cos nO cos (2fsin 0) dO, (n even), 

1 r* 

= - I cos ntf cos (z sin tf ) dtf , (n odd), 

1 f* 
J^ (fr) = - I sin 72^ ain (2? sin 0) dd, (n odd), 

1 r* 

= ~ I sin nd sin (z sin ^) dtf , (n even). 

TT Jo 

Since 

cos (ntf — e sin tf ) = cos nO cos (^ sin 0) + sin ntf sin (z sin tf), 

we have in all cases when n is an integer 

1 f* 
Jn (z)=i— I COS {nd — J? sin 0) d0, 

TT J Q 

the formula required. 

Example, To shew that for all values of n, real or oomplez, tbe integral 

1 fw 
y—- I ooe (n$ - $ mn $) dS 
ir J 



154] BESSEL FUNCTIONS. 279 

satisfies the differential equation 

^ ^ dy . f-, w'N sin nn /I n\ 

which reduces to BessePs equation when n is an integer. 

1 /■» 
Forif y = - I cos(n^-28ind)eW, 

wehave 7^^~ I sindsin (?i^-«sin^)cW, 



so 



cPv 1 r* 

-^= I sin*^cos(n^-«8in^)cW, 

p 

y+^ = - / cos*dcos(7id-«sind)cW, 



and " "J"^ ^y^~ \ sin(nd— 2Sind)cW — I pCos(nd-«smd)cW. 



Now integrating by parts, we have 



-I — — sin(n^-2sin^*e^«— sinnTT-l — I — (n-;^cos^)cos(w^-«sind)eW, 
vjo z nz vjo z ^ 



and therefore 



1 t A -- ^ 

«= — sinn7r--«- / cos(n^-«sind).rf(n^-«sind) 
nz «"«• y ^»o 

1 . n . 

sin?tir /l _»\ 
which is the required result. 

164. Extension of the integral-formula to the case in which n is not a/n 
integer. 

We shall now shew how the result 



1 f » 

(xr) = - I cos {nd — zsmff) dO 



must be generalised in order to meet the case in which n is not an integer, 
ie. the case of the Bessel functions^ as opposed to the Bessel coefficients. 



280 TRANSCENDENTAL FUNCTIONS. [CHAP. XII, 

Suppose that the real part of z is positive. Write ^ = ^ ru in the formula 



we thus have 






where u"*^^ has that value which becomes unity when the variable u travels 
by a rectilinear path to the point w = 1. Since values of t whose real part is 
large and negative correspond to values of u whose real part is large and 
negative, we see that the path in the u-plane, along which this integral is to 
be taken, is still a path leading from u = — x round the point u = and 
returning to w = — oo . 

Let this contour be chosen so as to consist of 

(a) a straight line parallel to, and below, but indeBnitely close to, the 
real axis from it = — oo to w = - 1 ; 

()8) a circle I of radius unity described round the origin ; 

(7) a straight line parallel to, and above, but indefinitely close to, the 
real axis from u = — 1 tot4 = — 00. 

Thus 



'^»<^> = 2^/_">-^'^^""-^'^« + 2^/,«-""^ 



1 /•-• ?U-^) 

ZTrt j-i 



where u~*"* has in the first integral the value «<*■♦■»)*» at u = — 1, and in the 
third integral has the value 6~<'^*^*' at u = — 1. Hence, writing w = — t in 
the first and third integrals, and u = e^ in the second integral, we have 



1 r* ^(»+i)t» /•• -^-f+-\ 



dt 






where, in the last two integrals, t"**~* has the value 1 at the point t«l. 
Writing t = «•, we have 

^ sin(n + l)7r r^_^.,^„h#^^ 

TT Jo 

^trjo ' TT Jo 



154] BES8EL FUNCTIONS. 281 



or Jn(z) = - ['coaizsine-neyde-''^^^'^ I e-»»-"inh«d^ (1). 

This formula is valid when the real part of £: is positive. When the real 
part of z is negative, a similar procedure leads to the result 

J^ (z) = ^— j r cos (z sin + n^) d0 - sin nir j e-«*+'»inh<> ^^1 (2). 

When n is an integer, the formula (1) gives 

1 f* 
J„(^) = - j cos (w^ — 2: sin ^) d^, 
'jr Jo 

when the real part of -^ is positive ; and the formula (2) gives 

J„ {z) = tllZ! r cos {ne + z sin 0) d0, 

or, since /« i^) = (- 1 )** J-^ {z\ 

1 f"" 
t/n (-2^) = - cos (nd — 2^ sin 0) d0, 

TT Jo 

when the real part of z is negative. 

Thus in either case when n is an integer, we have again the result of the 
last article, namely the formula 



'„(^)=1 rcos(/i^-2rsin^)(i^ (3). 



The equation (3) was kDown to Beasel. Equation (1) is due to Schlafli, Math. Ann. m. 
<1871) ; equation (2) was first given by Sonine, Math. Ann. xvl (1880). 

The trigonometric integral-formula for J^ (z) may be regarded as corresponding to the 
Laplacian definite integrals for the Legendre functions. For we have seen that the 
Bessel function J„^ (z) satisfies the differential equation of the function 



Limit P^ 

11—00 






or 



or 



But the Laplacian integral shews that this quantity is a multiple of 

^* I'ob "£"' "^ (0 " £')*" i}*««*J*=°"'^ '^ 

Limit I fl-i — cos<^] ooHfmf>d<l>f 



I ^OM^GOBin<f>cUl>j 

the similarity of which to the above result (3) will be observed. 



282 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 

166. A second expression of Jn {s) as a definite integral whose path of 
integration is real. 

Another definite-integral formula, which is valid for all values of z and a 
certain range of values of n, can be obtained in the following way. 

The function Jn (z) is expressed for all values of n and z by the series 

I (-l )r^+«r 



rro2'»-^r!r(n+r+l)* 
Since (§ 95) we have 






this can be written in the form 

(-l)-z"^r(r + l) 



Jniz)= S 



■0 2»r(n + r+l).2r!rQ) * 
Now by § 107 we have 

provided the real parts of (r + ^ ) ai^d (n -f = ) are positive. 

Thus if the real part of ( '^ + «) ^ positive, we have 

•^nW«-— TXr— 7 n 2 ^ — ^ cos^'V^sin'^i^d^. 

2«ryjr(w + ijr=o ^r\ Jo 

Tj ^ / .X ^ (-l)''^cos*'^ 

But cos (£^ cos <p) = 5) ^— -. , ^ . 

^ ^' r^^Q 2r! 

Thus we have . 

J (z) =s J, I J .1 I COS (2: COS 6) sin** ^ (i<5. 

2-rQ)r(n+y'o 

This formula is true for all values of z, and for all values of n whose real 
part is greater than - ^ . 

Example 1. Shew that 

P^ (cos 6) « jT^^fYj j^ e''<^» Jo (^ ffli^ ^) ^ dx. 



80 



155, 156] BESSEL FUNCTIONS. 283 

For we have 

fry 

Jo ir J J 

= r(n+l)P»(co8^), 
which establishes the result. 

Example 2. Shew that 

Pn"» (cos ^)= -7—4-7 fx f e''^9J„,(xamB)x^dx. 
r(n— m+i;y o 

(Cambridge Mathematical Tripos, Part 11^ 1893.) 

166. Hankers definite-integral solution of BesseVs differential equation. 

If in the result of the last article we write 

t = cos (f>, 
we obtain the result 

2"r(i)r(n-.y^-i 

It will now be shewn that this integral is a member of a very general 
class of definite integrals which satisfy Bessel's differential equation, namely, 
integrals of the form 

y=^n f e^ (t^ ^ l)n'h dt, 

Jo 

where C may be any one of a number of contours in the ^-plane. The 
importance of solutions of this type was first shewn by Hankel*. 

To shew that integrals of this class satisfy BessePs equation, we form the 
first and second derivates of the expression y, and find that 






= -z^' ( {^e«'((»- !)"+»- (2n + l)He^ («•- I)""*} 
J c 



dt 



* Math. Ann. i. 



284 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 

From this it is clear that Bessels equation will be satisfied by the integral 

J c 

provided is a closed contour such that the integrand resumes its initial 
value after making a circuit of C 

The similarity of this result to the general theorem of § 142 is very 
apparent. 

167. Expression of Jn (z), for all vaiues of n and z, by an integral of 
HankeVs type. 

We shall now shew how the particular solution Jn {z) of Bessel's equation 
can be expressed by an integral of HankeFs type. Consider the contour 
formed by a figure-of-eight in the ^plane, enclosing the point ^ = + 1 in one 
loop and the point ^ = — 1 in the other, so that a description of the contour 
in the positive sense involves a turn in the positive direction round the point 
< = + 1 and a turn in the negative direction round the point ^ = — 1. After 
turning round the point < = + 1 in the positive sense, the integrand resumes 
its original value multiplied by ^»-*>*'*, as can be seen by writing it in the 
form 

««+(n-i)log«-l)+(i»-4)log(e+l). 
^ » 

and after turning round ^ = - 1 in the negative sense, it is further multi- 
plied by 

-(i»-i)fcrt 

Hence after describing the whole contour, the integrand resumes its 
original value. 

rd+.-i-) 

Thus y^i^\ e**(e«-i)»-*(ie 

is a solution of the differential equation, valid for all values of z and of n ; 
the symbol (1 +, — 1 — ) placed at the upper limit of the integral indicating 
that the path of integration consists of a positive revolution round 1 and a 
negative revolution round — 1 

In this equation we shall suppose as usual that ^ has the value which 
reduces to I when z travels by a straight path to the point z^\, and we shall 
suppose (^" — I)*"* to have initially the value which reduces to «-<*-*)'» when 
t travels by a straight path to the point ^ = 0. 

To find the relation between this quantity y and the particular solution 
Jn {z) of Bessel's equation, we expand y in the form 



157] BESSEL FUNCTIONS. 285 

To evaluate the iutegrals which occur in this series, write 



F{r,n)^j 



(1+.-1-) 



m+,-1-) 
Then F(r, n+ l) = j (r+«-r)(<«- !)»-*(£« 



a+.-i-) ^H-i 



-I 

/•(i+.-i-)(««_l)«.+i(r + l)^- „, . 



Thus we have F{r, n) = - ^ ^"T" jP(r, n + 1). 

This result enables us to reduce the evaluation of F(r, n) to the evalua- 
tion of F(r, n + 1), and thus to the evaluation of F(r, n + k), where A is a 

positive integer so chosen that the real part of (n + A:) is greater than — ^ • 
We have therefore to evaluate the integral 



F(r, n)=|' 



where we may now suppose that the real part of n is greater than - ^ . The 

contour can be supposed to start at the point t = 0, where (^'— I)**"* has the 
value «-<'»-*>»», then to proceed to the neighbourhood of the point ^ = 1 along 
the real axis, then to make a positive turn in a small circle round t = 1, then 
to return along the real axis to the point ^ = 0, where (^*— 1)**"* has now 
the value e<»-*)'^, then to proceed along the real axis to the neighbourhood 
of the point ^ = — 1, then to make a negative turn in a small circle round 
^ = — 1, and lastly to return along the real axis to the point t = 0, where 
(^— l)**"* has now the value e"^""*^**. Since the real part of n is greater 

than — 2 , the integrals round the small circles at ^ = 1 and ^ = — 1 are 
infioitesimal, and we therefore have 



^0 Jo 



dt 



JO Jo 

where in each of these integrals the quantity (1 — <■)*"* is now supposed to 



286 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 

have the value unity at ^ = 0. Writing — ^ for ^ in the two last integrals, 
we have 

F(r, n)= - {e^"-*)** - «-<»-*)'<} {1 - (- 1)^-1} JV(1 - t*)^-^dt 

= — 2% cos* -x-sin(n— sJtt/ v* <^"*^ (1 — i/)*-* dv, where v = <", 
<by § 105) = - 2i C08' ^ sin (n - 1) ^fi (^ , n + ^) 

(by §106) =-2ico8'^sin(n-l)^ ) r \^' ' 

r^n+2 + l) 

This result has now been proved to hold so long as the real part of n is 
greater than — = : and in virtue of the formula 

F(r. n) g^^^ Fir.n + 1), 

we see that it holds universally. 

Thus we have F(r, n) = 0, when r is odd ; and it is therefore sufficient to 
take r even. Let r = 2s. Then the formula becomes 

^•(2,, n) = 2i sin (» + 1) ^ \^/; \ , V . 
But (§ 97) we have 

r(„+l)rQ-„) — 



TT 



sm 



Therefore 2^V^/'« + l^ 

/'(2», n)=— ^^ -_i U . 

r(i-n)r(n + , + l) 

and so y = 2 ^^ — a-, M < 

'"> 2«! r (i-n)r(n + «+l) 



or 



,^ » (-l)^+» ^^"^^ (§) 

^ .=« 2-«! r(i-n)r(» + ,+ i)- 



But 



"^ ' ,ro2-+»«!r(n + «+l)* 



158] B£SSBL FUNCTIONS. 287 



Therefore J„(.)=_^-^,. 



(ly 



This formula gives the required expression of J^ {z\ It is valid for all 
values of n and of z\ but when n is of the form (^ + 2)» where A is a 
positive integer, the factor T (o — w ) becomes infinite and the integral 



/ 






becomes zero (since the integrand is now regular at all points within the 
contour), so that for this exceptional case the formula is indeterminate. 

Example, Deduce the formula 

from the result of this article. 

« 

168. Bessel functions as a limiting case of Legendre functions. 

We have already (§148) shewn that Bessel's differential equation of order 
771 is the same as the differential equation of the associated Legendre 
functions 

We shall now express this connexion more precisely, by establishing the 
formula 

^„ (.)= Limit n-.P,«.(l-^.). 

For taking the expression of the associated Legendre function by a definite 
integral (§ 131), we have 

~mpm(-t _ ^ (n-\-m){n + m-\) (n-TO + 1)^" / ^N** 

" " V 2»»y" (2to - 1) (2m - 3) l.-n-.n"" \ 4m*} 



/„'{^-£« + '^*(-| + £')P«^""^'''^' 



and as n becomes infinitely great, the right-hand side of this equation tends 
to the limiting value 

7^5 rr-To ^r = / ( 1 + - COS ^) sin"*Ad<f>, 

(2m — l)(2m — 3)...l .TrJo \ n v ^ ^' 



288 TRANSCENDENTAL FUNCTIONS. [CHAP. XU. 



— o)... 1 . TT Jo 



(27?i-l)(2m-3) 

or (§165) Jn.{zy> 

which establishes the result stated ; it is due to Heine*. 

169. Bessel functions whose order is half an odd integer. 

The result of § 157 suggests that when the order n of a Bessel function 

Jn (z) is a number of the form A? + 5 , where A? is a positive integer, certain 

exceptional circumstances arisef in connexion with the function. In this 
case it is in fact possible to express the Bessel function 

•^*^^^^"2*+*r()fc + ?) I 2(2A: + 3)'*'2.4.(2A; + 3)(2A + 5) ••*) 

in terms of well-known elementary functions. 

For by § 161 we have, if A; be a positive integer, 

/„«-(-2)..«4^|-'.W}. 



But the series-expansion of the function J^ {z) is 



2\i . 

sin^r. 



. , , 2M f, z* z* ) / 2 \ 

Therefore J,^i {z) -^ ^ ^^,^ (-^J , 

which is the required expression of the function Jj^ {z) in terms of more 
elementary functions. 

The student will without difficulty be able to prove that a second solution of Bessel's 
differential equation in this case is 



* Heine's definition of the associated Legendre function is somewhat different from that 
which has since become general and which is adopted in this book : this leads to differences of 
statement in many other formulae, such as that of this article. 

t The student who is familiar with the theory of linear differential equations will observe that 
in this case, and also in the other exceptional case of n an integer, the difference of the roots of 
the '* indicial equation *' of BessePs equation is an integer. 



159, 160] BESSBL FUNCTIONS. 28& 

Example. Shew that the solution of the equation 

IS y-« 4 2 Cp{*^-m-* (2apl*)+»J'« + i(2a^)}, 

p»0 

where the quantities Cp are arbitrary constants, and o^y a|, ... o^mi &x^ ^^ roots of the 
equation 

o***^— i (LommeL) 

160. Expression of Jn (z) in a form which furnishes an approximate value 
to Jn {z)for large real positive volumes of z. 

We now proceed to form an integral which will be found to play the 
same part in the theory of the function J^ {z) as the integral of § 104 plays 
in the theory of the function T {z). We shall suppose 5 to be real and 
positive. Then, by § 155, we have, for all positive values of n, 

J^ (z) SB / cos {z cos <f>) siD^<f> cUf>, 

Writing cos <^ = a?, this becomes 

Jn(z)=^ J 11 I (1 - a^y^ COS zxdx, 

or Jn(-^) = Real part of /^ . /^(l - x')'^e^dx. 

In order to transform this integral, we take in the plane of a complex 
variable t a contour OPQBCOy formed in the following way. is the origin 
(^ = 0) ; P is the point ^ = 1 — />, where /> is a small quantity, and OP is the 
part of the real axis between and P. Q is the point ^ = 1 + 1/), and PQ is a 
quadrant of a circle which has its centre at the point ^ = 1. B ia the point 
t^l -hik, where A; is a large positive quantity, and QB is the line (parallel to 
the imaginary axis in the ^plane) joining Q and B, C is the point t = ik, and 
BC is the line (pcurallel to the real axis) joining B and C, Lastly, CO is the 
part of the imaginary axis between C and Q. Then the function 

is regular at all points of the ^plane in the interior of the contour OPQBCO ; 
and therefore the integral 

taken round this contour, is zero. 

w. A. 19 



290 TRANSCENDENTAL FUNCTIONS. [CHAP. XIL 

We can write this relation in the form 



f +f +f +f +f =0. 

J OP J PQ JOB JbC J CO 



Now the part of the integral due to PQ tends to zero with p, and the part 
due to BC tends to zero as k becomes infinitely great, while the part due to 
CO is purely imaginary. Thus we have 



Real part of / = — Real part of I , 

J OP J QB 



and so Jn W= Real part of , ^ ,i f (1 -fy*^^cU. 



In this integral write 



%u 
z 



80 that u varies between the limits and oo when t describes the line QB; 
and (i_^)* = /f(?y«*(l + gy. 

and therefore f =2"-*w {"*'""^r-»-*rr^M-* (l + g)*~*dM. 
Thus we have 



or j; («) = 



r(n+i).(27r^)» 

'-{-(-i)i}/>»-i(>4r-('-ir}*'' 

-"{'-(«4)f}/:— {(>-ir-('-£n<<'«j 

This is the integral-expression required. It is easily seen to furnish an 
approximate value of J^ (z) for large positive values of ^ ; for as ;? becomes 
indefinitely large, the two integrals in the expression tend respectively to the 

limits 2r (n + ^ j and zero ; and therefore the function Jn (z) approximates 
for large positive values of z to the value 



(l;)*^|'-KI)l}- 



160] BESSEL FUNCTIONS. 291 

The evaluation of «/» (z) when z is large will be considered in fuller detail 
in the next article. 

The result of this article can also be obtained in the following quite different manner, 
which connects it more closely with the general theory. We have seen in § 148 that Bessel's 
differential equation is a limiting case of the general hypergeometric equation, represented 
by the function 

00 e 



n i-2tc 2w?-l 

Since the differential equation of the P-fimction 

(0 00 c 
a p y Z 

is (§ 142) satisfied by the integral 

taken between suitable limits, we see that Bessel's equation is satisfied by the expression 

Limit «<•«-* (<»-» (l-^y •-»+«• («-0*'*<^, 

or ««•«-* [ r-J «-«< {z - <)•"* dt, 

or (putting < «= - ivz) 

«<•«-» jif''-hs^'k{z+%vz)*-he-^zdv, 



or «*V» / (v+tv«)*-*e->"cfo. 



The limits of the integral can be taken to be and oo , since these satisfy the conditions 
for the limits found in § 142 ; and hence it follows that 

is a solution of Beesel's equation. 
Similarly the quantity 

is a solution of Bessel's equation. 

The solution J^ {z) must therefore be of the form 

J^(z)»A^i^ r {v+iv^'ie'^dv'¥Be'*'s^ riv-iv^'^'ie'^dv, 

19—2 



292 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 

where A and B are constants independent of i. This is substantially the form given above, 
but the determination of the constants A and B \b a, matter of some difficulty, for which 
the student is referred to a memoir by Schafheitlin, CrelU^s Journal^ oxiv. p. 39. 

Example, Shew (by making the substitution u^2zcot<t>m the integral found above^ 
or otherwise) that 

ir 

J^{s)^ ?^^^^?!^. fV««~**oos«-*<^oosec*»+i<^cos{«-.(n-i)^}c&^. 

r(n+i).v*Jo 

161. The Asymptotic Expansion of the Bessel functions. 

The Bessel functions can for large values of the argument be represented 
by asymptotic expansions. We shall here consider only the asymptotic 
expansion of Jn{^) for positive real values of z; this was discovered by 
Poisson (for n = 0) and Jacobi (for general integer values of n). The theorem 
has been considered for complex values of z by EUmkel * and several subse- 
quent writers. 

We shall derive the asymptotic expansion from the integral-expression 



Jn(z) = 



(iwz)^ r (n + 1) 



■-('-(- 1)1} /:--'i('4r-(' -sn^» 

found in the last article. 

It is first necessary to find the asymptotic expansion of the integral 

which we shall denote by the symbol /. 
Now we have 



iu 



^.jt2Li^)r(g..)-(>,.^.^. 



* Math, Ann, i. 



161] BESSEL FUNCTIONS. 293 

Therefore 



tu 



^ Mt-i)^..(t-.) jv^.._/;(g-.)-(i^.)^.^, 





or 



where 

Now as z becomes infinitely large, n having any definite finite integer 
value, the remainder-term R^^ tends to the limit 

It follows from this that 

Limit z^R^ = 0, 

and therefore the series 

r(ifc+ 1) jlH- i^(A^ + r)(^-Hr-l)...(fc-r + l)0| 

is the asymptotic expansion of the function 



r e^u^fl + ^^j'du (k>0). 



Substituting this result in the expression already found for «/,(-?), we 
see that 



rU^) 



(2,r^)* r(n + i) 



^r(" + 2)2tf + i. W (fe^l 

^- i f ■ l\-^)f^ (""^•^'')("~^+'')-("~^-^5)i'-^.-,-.(-t)r | 



COS 



294 TRANSCENDENTAL FUNCTIONS. [CHAP. XIL 

is tbe asymptotic expansion of the Bessel function «/n(^) for large positive 
values of z. 

Even when z is not very large, the value of c/» {z) can be computed with great accuracy 
from this formula. Thus for all values of z greater than 8, the first three terms of this 
asymptotic expansion give the value of Jq (z) and Ji (z) correct to six places of decimals. 

162. The second solution ofBesseVa equation when the order is an integer. 

We have seen in § 149 that when the order n of Bessel's differential 
equation is not an integer, the general solution of tbe equation is 

where a and fi are arbitrary constants. 

When however n is an integer, we have seen that 

and consequently the two solutions Jn (*) and «/_n (z) are not really distinct. 
We therefore require in this case to find another particular solution of the 
differential equation, distinct from Jn(^)t in order to have the general 
solution. 

To obtain this second solution, we write 

y = uJn (z), 

where u is a new dependent variable, in Bessel's equation 



Remembering that Jn(^) is a solution of Bessel's equation, the differ- 
ential equation for u becomes 



162] BESSEL FUNCTIONS. 296 

dj^ a dz . 1 rt 

or -_ + 2-y— — + -=rO. 

dz 
Integrating this equation, we have 

UtUt 

log ;j- + 2 log Jn {z) -h log z = constant, 

dvt 6 . « • 

or -=- = . - . .. , where 6 is a constant, 

dz z [Jn {z)Y 

where a and 6 are arbitrary constants. 

The complete solution of Bessel's equation can therefore be written in 
the form 

To find the nature of the solution thus obtained, we observe that in the 
vicinity of the point ^ = the integrand 

is of the form 

^~**~* (constant + powers of f)~*, 

which when n is a positive integer can be expanded as a Laurent series in 
the form 

The function 

l'tr'[Jn{t)]''dt 
has therefore the form 



where the quantities d^^, ^^-m+s* ••• ^^^ definite constants. 

It thus appears that the complete solution of fiessel's equation can be 
written in the form 

y = ilJn(^) + 5{/»(^)log^ + t;}, 

where v is the result obtained by multipl3nng together J^{z) and a Laurent 
series of the form 



296 TRANSCENDENTAL FUNCTIONS. [CHAP. XIL 

where the quantities (£-an> <2-sn+af ••• ^^^ definite constants, and A and B are 
arbitrary constants. The expansion of Jn (^) being known, we see that the 
product V has the form 

jt^ X a power-series in ii* ; 

and thus a second solution of BesseVs differential eqiwiion, in the case in 
which n is an integer, can be taken of the form 

Jn (z) log 2r + 2r« (Oo + OiZ* + Oi-e* + a,;^ -h ...), 

where the quantities o^, Oi, a,, ... are definite constants. These quantities a 
are not however all of them strictly speaking definite, since by adding a 
multiple o{ Jn{z) (which will leave the expression still a solution of Bessel's 
equation), it is possible to change all the quantities a after On-i. 

This solution will be denoted by Kn{z)*. 

The coefficients Oq, Oi, a,, ... may theoretically be determined by substi- 
tuting this expansion in the differential equation, and equating to zero the 
coefficients of successive powers of z, A better method is however the 
followingf. 

We have seen that when n is a positive integer, J.^(z) reduces to 
(— l)^Jn{z); in fact, if in the equation 

T ( \ l^X'"'^^ T (-1)^ /^\^ 

J-(«-.,W=^2^ p:or(-n + 6+p-hl)r(2> + l)V2; 

we suppose the quantity € to tend to zero, all the terms of the series vanish 
as far as p = n, since r(— w-hp + l) is for these terms infinite. Changing 
the meaning of the index of summation /> in the other terms, we have 

./_(„_, w W p.«r(-n+e+;)+i)r(p+i)Uy 



^ ' \v ,tor(€+p + i)r(n+p + i)V2y ' 



and when e = 0, the first of these partial series is zero and the second is 

(-l)»J„(e). 

Since the quantity 

(-1)»J_,^,(^)-J,„_,(^) 

vanishes with e, we can take as a second solution of BesseVs equation the 
limiting value of the quotient 



* In referring to memoirs it most be borne in mind that different writers have taken different 
definitions of the Besael fonctions of the second kind, 
t Dae to Hankel, Math, Ann, x. p. 470 (1869). 



162] BESSEL FUKCnONS. 297 

Substituting the above values for the Bessel functions, this becomes 

\2) ^toT(p + l)€T{-n + e + p+l)\2) '*'V2/'pto^ ^ e \2) ' 
where /(€) represents the expression 

^^^^"r(n+p + i)r(p + 6+i)(2J "r(n+p-€+i)r(p+i)V2J ' 

The limiting value of /(€)/e, as € tends to zero, is 

^ 21og^ ^ ^<^ + ^> 

1 r(«+p+i) 
r(p+i){r(n+p +!)}•• 

Also, since 

we have limit ,r(_„^^,^^^i) = (- l)"^' r (« - p). 

Consequently we obtain, as a second particular solution of Bessel's 
equation, the expression* 

\2; ptor(p+i)V2; ^{2) pZ,,r(n+p+i)T{p+i)\2) 



z r(n+p + l) F(p + 1)] 

r"'*2 r(n + p + i) r(p + i)j* 

The coefficient of log« in this expression is 2J,(«). So, dividing the 
expression by 2, we have tbe second solution in the form ' 

j.(«)i»g.-^;i;E^(i)''-^.wiog2 



+ 



(iXi. (-i)p cy^l ^(n+p+ i) r(p+i )] 

W pr«r(n+p + l)r(j) + l)V2/ 2 1 T(n+p + l) r(p + l)j 



It is convenient to add to this e^cpression a term 



/,(.){log2+M)|. 



* This 18 Bankers seoond solution Y^ (z). It is really 

dn ^ ' dn ' 



298 TRANSCENDENTAL FUNCTIONS. [CHAP. XU. 

which is itself a solution of Bessel's equation ; so the second solution now 
takes the form 

^n(.)iog.-2^2— |j-^y 

^\i^S\ (- \y /^\^ [F (n -h p + 1) r (p +1) 2r (i) ] 

2Wptor(n+p+i)r(i)+i)V2>' lr(^+i) + iK r(p+i) r(i) r 

This is the solution K^ {z) which we take as our standard. 

Since, when r is a positive integer, we have 

r(r + l) F(l)_ 11 1 

r(r+l) r(l) '■^2'^3'^"'^r' 

we can write K^{z) in the form 

_|(|ri^z^j,+ui+...+i+i+|+...+ i_i(|r. 

2\2/ ^«o (»H-l>)lpl I 2 3 p 2 7i+pJ V2/ 

When n is an integer, the two independent solutions of Bessel's differ- 
ential equation are J^ {z) and K^ (z\ 



Example 1. Shew that the function K^ (z) satisfies the recurrence-formulae 

These are the same as the recurrence-formulae satisfied by J^ («). 

Example 2. When the real part of 2 is positive, shew that the expression 

r8in(isin<^-n<^)ci(^- j e-*«toh«{e««+(-l)*e-«^ dS 

is a second solution of Bessel's differential equation of integer order n. 

(SchlaflL) 

Example 3. Shew that the expression 

yologi+2(y,-^4+iJ5-...) 
is a second solution of the Bessel equation of order zero. 



163] BESSEL FUNCTIONS. 299 

163. Nexmianns expansion ; determination of the coefficients. 

We shall now consider* the expansion of an arbitrary function f(z), 
regular at the origin, in a series of Bessel functions, in the form 

f(z) = OqJo (z) + a^Ji (z) + aa/2 {^) + " •» 

where the coefficients a©, «!, ««> ••• are independent o{ z. , 

Suppose first that such an expansion is possible, and let us try to 
determine the coefficients, by expanding both sides of the equation as 
power-series in z and equating coefficients of the several powers of z. Since 

/(^)=/(0)+2(|)/'(0) + |(|)V"(0)+|^(|)V"'(0) + ... 

and «^»W = ^(|) {l-i!(„Vi)(|) +2!(n + lKn + 2)(l) --}' 
we have on comparing coefficients the equalities 

/(O) = «.. 
2/' (0) = a„ 
2"/" (0) = - 2a. + Oa, etc., 
from \rhich without difficulty we find 
a. = /(O). 

«„ = 2 1/(0) + JV" (0) + "'^"4;"^'^ /'^ (0) + . . . + 2»-»/ <«) (0)| (n even). 

«„ = 2|n/'(0)+"-(^^V"(0) 

+ »(»'-l')("'-3') ^,T) (0) + . . . + 2»-'/ <») (0)1 (n odd). 

These coefficients take a simpler form, if we introduce functions Oi {z), 
Ot (e), 0, (z), ..., defined by the formulae 

^,. » n(n*-V) n (n» - !•) (n» - 3") 2»-in! . ,,, 

for then it is easily seen that On is twice the residue of the function On(t)f(t) 

* 0. Nemnann, Theorie der Bet$eV$ehen Functionen. The exposition here given followB 
Eftpteyn, AmuUet de Vkeolt Normate (8) z. p. 106 (1898). 



300 TRANSCENDENTAL FUNCTIONS. [CHAP. XH. 

at the point < =» 0. The two formulae for On (^) can be united by reversing 
the order of the terms ; thus 

OnW«-^jrH-|l + 2(2n-2)"^2.4(2n-2)(2n-4)"*"-J' 
the series terminating with the term in z^ or jf^K 
We thus have Neumann's expansion 

f(z) = Oo/o W + «i/i (z) + a, J, (z) + ..., 
where ao=/(0)/ 

and On (n > 0) is twice the residue of 0» {t)f{t) at the point ^ = 0, so that 



On^^^l On(t)f{t)dt, 



y 
where 7 is any simple contour surrounding the origin. 

164. Proof of Neumanria expansion. 

The method by which this result has been found cannot be regarded as a 
proof, since the possibility of the expansion was assumed. We can, however, 
now furnish a proof by determining directly the sum of the series obtained. 

From the definition of On {z\ we can at once obtain the identities 

0„+. {z) + 2 ^^ - 0„_, {z) = 0, (n > 0). 

d tV 



".W'-s©. 



Writing the first of these equations in the symbolic form 

On+x-2DOn-On.i-0, where i)-^, 

and solving the series of recurrence-equations obtained by giving n integer 
values, in the same way as if D were an algebraic quantity, we obtain for On 
the symbolic expression 

On(^) = i[{-i) + (i>+l)»}-+{-i)-(i>'+im(j)- 

This symbolic expression can be transformed into a definite integral in 
the following way. 



164] BESSEL FUNCTIONa 801 

1 r* 

We have - - / e~*^ du, 

t Jo 

where the upper limit must be understood to mean that direction at infinity 
which makes the real part of tu positive and infinite ; and therefore 

or, writing tu = a?, 

On (t) = f *i tr^' e'^[[w + (^ + ^«)*}« + {x^(a^ + <»)*}«] dw, 
Jo 

where the upper limit now means the real positive infinity, so that the 
integration may be regarded as taken along the real axis of x. 

Writing this in the form 

<'-<«-r?rj:[r-i^T^<-'>-{^T(^/]-'^' 

we have 0, (<) J, («) + 2 2 On (<) J,, («) 

(by § 146) 
« ^ Limit I 2^ < * xH»^+pM . e"* dx 

^t x«oo Jo 



y Limit / 

t JT.flo Jo 



^ — -« 
6* " cia?. 



In order that this integral may have a meaning, the real part of —r— 

V 

must be negative, a condition which is fulfilled when 



If this inequality is satisfied, we have therefore 

Oo{t)Jo(z) + 2 X On{t)Jniz) = :r^ . 

« 

From this result Neumann's expansion can at once be derived ; for let 
/(z) be any function which is regulai* in the interior of a circle C whose 
centre is at the origin, and let ^ be a point on the circumference of the circle. 
Then if z be any point in the interior of the circle, the condition | jp | < | ^ | is 
satisfied, and therefore we have 

^ ^Oo(t)Jo(z) + 2iOn(t)Jn(z). 



t-Z 



n«l 



302 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 



Thus /(^) = ^— . y^— 
•^ ^ 2'fnJc t — z 



= 2^|^/(e)d^ jOo (0 Jo (z) + 2X0n{t) Jn (^)} 

where flo =/(0) 

and On = — . f On (0/(0 ^^ (^ > <>). 

This establishes the validity of NeumaDn's expansion for points z within 
the circle C, 

Example, Shew that 

cos f-^o W- 2/, (i)+2/4 («)-..., 

166. Schldmilch*8 expansion of an arbitrary fwnction in terms of Bessel 
functions of order zero. 

Schlomilch * has given an expansion of a quite different character to that 
of Neumann. His result may be stated thus : 

Any function f(z) which is finite and continuous for real values of z 
between the limits z=^0 and z^ir, both indu^sive, muy be expressed in the form 

f(z)^ao + Oi Jo (-8^) + Oi Jo (2-er) + a^Jo (3-8^) + . . ., 

where Oo = / (0) + - f ' w f \l - ^)-*/' (ut) dt du, 

^Jo do 

an^- I ucosnu I (1 - ^*)~*/' (ut) dtdu (n > 0). 

^Jo Jo 

Schl5milch's proof is substantially as follows 

Suppose that F and /are two functions connected by the relation 

f(z) = lf\l-^)-iF(zs)ds. 



Then we have 

2 p 





f(z) = -[\l-'^)-^sr(zs)ds. 

TTJo 



♦ ZeiUehnftflT Math. u. Phynh, n. (1S67). 



165] BESSEL FUNCTIONa 803 

In this equation, write zt for z, multiply both sides by -er (1 — ^«)-* dt, and 
integrate with respect to t between the limits ^ = and t^\. Thus 

^ ['(1 - t^yif(zt)dt - — (\l - «•)-* dt l\l - l^)'^8F'(Z8t)d8 
Jo IT J Jo 



2 r» /•(*•-»•)* 

where x^zst^ y^z8{l — t^)^. 

Performing the integrations, we have 



-/ {s^-a^''fy^F'{x)dxdy, 

TT J Jo 



f \l - e«)*/' (^) (ft = F{z) - J?'(0). 
Jo 



2: , 
'o 



Now by the definition of the function/, we have 

/(0)^F(0). 

Thus F{z) =/(0) + z f\l - <«)-*/' (zt) dt. 

Jo 

This equation expresses the function F explicitly in terms of the function 
/, whereas in the original definition / was expressed explicitly in terms 
o{ F. 

In order to obtain Schlomilch's expansion, it is merely necessary to apply 
Fourier's theorem to the function F(zs). We thus have 

f(z)^- ( (l-«»)-*(fojl ('F(u)du + - 1 rcosnwcosn^m)^^! 

TT^o {jrJo ^ i»«lJo j 

If' 2 * T' 

= - I F(u)du + — 2 / cos nuF{u) Jo (nz)diu 
^Jo ^ »»i Jo 

In this equation, replace F{u) by its value in terms oi f{v). Thus 
we have 

+ - i J, (tl?) r cos nw 1/(0) + u f (1 -«•)-*/' (we) (ftl dw, 
^ »-i Jo I Jo ) 

which is Schlomilch's expansion. 

Example, Shew that if < f < ir, the expression 

|*-2 |^,(*) +5^o(3»)+i^.(6*)+...} 



804 TRANSCENDENTAL FUNCnON& [OHAP. XIL 

is equal to z ; but that, if ir ^f ^ 2ir, its value is 



«+2irC08-i--2(*«-ir*), 



where cos'^ - is taken between and -x . 
z 3 



Find the value of the expression when » lies between 2«r and Zw. 

(Cambridge Mathematical Tripos.) 

166. TainJ^iXtion of the Bessel functions. 

Many numerical tables of the Bessel functions have been published. 
Meissel's tables (Berlin, 1889) give the functions Jo(z) and Ji{z) to 12 
decimal places for real values of z from jer = to er = 15^, at intervals of 001. 

Tables of the second solution F© (z), defined by the equation 

z^ I 1\ ^ 
Y,{z)^J^{z)\o%Z'\r-^^''\\ + 27 2rT« ■*■•••' 

from ^ s= to ir = 10*2, are given by B. A. Smith, Messenger of Math, xxvL 
(1897). 

The British Association Reports for 1889, 1893, 1896, contain tables of 
the functions /n(^)> which are solutions of the differential equation 

dhb \du f. ^n^ ^ 

so that In(^)^i'^Jn(i^)' 

A table of the first 40 roots of Jo (^) is given by Wilson and Peirce, BtM, 
Amer. Math. Soc. in. (1897). 



Miscellaneous Examples. 

1. Shew (ag. by multiplying the expansions for e^ ~<^ and «~^ " if , and equating 
the terms independent of t) that 

{^oWl'+S {^1 (*)}»+2 {y, «}»+2 {J, (»)}»+... = !, 

and hence that, for real values of s, J^ {z) can never exceed unity, and the other Bessel 
coefficients of higher order can never exceed 2~l. 

2. Shew that, for all values of fi and v, 



MISC. EXS.] BES8EL FUNCTIONS. 305 

3. Shew that 



4. Shew that 



«^nW w+1- n-h2- w+3-../ 



5. Shew that 



/_mW«/m-iW+«^-m+iW*^mW== 



2sin/i9r 



«-« 



6. If v^-/! be denoted by Q^ («), shew that 



(Lommel.) 



dz z 
7. Shew that 



^_%ii).l_i(!LhI)e.w+,{e.W}«. 



8. If the function 



— I 2^co8'^t«cos(mu~;;sint«)(;?ti 



JJ{z)= 2_ :^,(i«)'»^-m.*.p, 



(which when k is zero reduces to a Bessel Amotion) be denoted by J^* (z\ shew that 

where N^m, *. p is the " Cauch/s number ** defined by the equation 

Shew further that this function satisfies the equations 
and ^m'^\z)=2mJ^' (z)-^2(k+l){jLi{z)-j!^+i{z)}, 



(Bourlet) 



9. If quantities v and i/'are connected by the equations 

Af^E-eainEy cosi;=_ jz. where |6|<1, 

«hewthat v=jr-h2(l-fl8)i 2 i (i«)»/^*(m«)-sinmJ/; 

-where J^*(^)=- I (2costi)*cos(mti-;?8ini«)(;?t^ 

"■y 



10. Prove that 



P,-(cos^)=^^/^|(a.*+y«)i ^1^, 



•where a«rcos^, ^+y*»»r*sin* $, and c„*» is a numerical quantity. 

(Cambridge Mathematical Tripos, Part II, 189a) 
W. A. 20 



306 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 

11. Shew that, if n is a positive integer and (m+2n+ 1) is positive, 

(Cambridge Mathematical Tripos, Part I, 1899.) 



12. Prove that 



2 r* 

Jo (2)=- I sin (« cosh «) fl?«. 



(Cambridge Mathematical Tripos, Part II, 1893.) 

13. Prove that 

and if Y^ (z) is HankeFs second solution of Bessel's equation, defined by the equation 

i r, (.) = Limit '^-.(^)-'^«(^)<=<^^ , 
T »=faitefer sm nir 

■^••^ ?'--»-i=-5i?<Mi,('*£r'(^')- 

14. Shew how to express ^J^ (z) in the form 

AJ^ z)-{-BJo{z\ 
where A, B are polynomials in z ; and prove that 

J4(6*)+3^o(6*)-0, 

3./fl(30*) + 6^,(30*)=0. 

(Cambridge Mathematical Tripos, Part II, 1896.) 

16. Prove that, if •/„ (a{)=0 and J^ 03ft=O, 

^^^xJ^{QX)M^x)dx^O, and |^*^{*^H(aa.0}«d:r-if8{^^^^(^)j8. 

Hence prove that the roots of J^{x)=0^ other than zero, are all real and unequal. 

(Cambridge Mathematical Tripos, Part I, 1893.> 
16. Shew that 

/* ^V~2~/ 

a:-*»+"»J«(ar)(ir=2-*+'«a'»-"»-i— --^— ^— , 

if 2w + l>f/i^-l. 

(Cambridge Mathematical Tripos, Part I, 1898.). 



MISC. EXS.] BESSEL FUNCTIONS. 307 

17. Shew that 



!- i '^{^^w 



n pao 



2 ^"ICiJ 



(Lommel.) 



18. Shew that the solution of the differential equation 
where ^ and ^ are arbitrary functions of «, is 



y-(^y<^-^^(^)+^-^-'W}- 



19. Shew that 



^^y /^V^(«sin^)sin'*+i^cW«2-*y«+j(«). 
20. In the equation 

the quantity n is real ; shew that a solution is given by 

( - 1)"» z*^ cos {u^-n log z) 



(Hobson.) 



cos (n log «) - S 



«-i 2»*m!(l+n2)*(4+n2)* (wi^+n^)* * 

where t^^ denotes 

tan-i?+tan-i5+...+tan-i-. 
12 m 

(Cambridge Mathematical Tripos, Part II, 1694.) 
21. Prove that the complete primitive of the differential equation 

where m is a positive integer, is 

u^AI^{z)+BK^{z\ 
where, for real values of z^ 

^" ^=1-73^^1) /." *-«-'"*«'>l>'"*'^- 

Prove also that 

/eo 
(««+««)-'»-Jco8Md«. 

20—2 



308 TRANSCENDENTAL FUNCTIONS. [CHAP. Xn. 

Shew that for very small values of z, 

jro(;^)--log|--677..., 
and that for very large values of z, 

(Cambridge Mathematical Tripoe, Part II, 1898.) 
22. If C be any ciu*ve in the complex domain, and m and n are integers, shew that 

j^O^{z)0^(z)dz^O, 

where k^O \f the curve does not include the origin ; and, if the curve does include the 
origin, 

it=0 if m + w, 

its=2»rt if m=n. 



CHAPTER XIIL 
Applications to the Equations of Mathematical Physics. 

167. Introduction : illustration of the general method. 

The functions which have been introduced in the three preceding 
chapters are of very great importance in the applications of mathematics to 
physical investigations. Such applications are outside the province of this 
book ; but most of them depend essentially on one underlying circumstance, 
namely that by means of these functions it is possible to construct series 
which satisfy certain partial differential equations, known as the partial 
differential equations of mathematical physics; and in this chapter it is 
proposed to explain and illustrate this fundamental property. 

The general method may be explained by considering first the solution 
of the partial differential equation 

^■^dj^=^ <1>' 

a solution which, while resting on the same principles as those to be 
developed later, does not require the use of any but the elementary functions 
of analysis. 

Consider any solution V{x, y) of this equation (1). Near any point at 
which a branch of the function V{Xy y) is a regular function ctf x and y, and 
which we may without loss of generality take as origin of coordinates, this 
branch of the function V{x, y) can by Taylor's Theorem be expanded as a 
power-series of the form 

^(^» y) = ao+ Oifl;-!- 6iy -h o,^ -h 6a^ 4- Cay* + Os^:* + ; 

on substituting this value of V in equation (1), and equating to zero the 
coeflScients of the various powers of x and y, we obtain the relations 

Oa + c, = 0, 

3a8 + Cs = 0, 
Sdj + 6, = 0, 



310 TRANSCENDENTAL FUNCTIONS. ' [CHAP. XIII. 

Fixing our attention on those terms in V which are homogeneous of the 
nth degree in x and y combined, it is clear that the equalities just written 
will furnish (n — 1) relations between the (n + 1) coefficients of these terms 
of degree n. When these equations are satisfied, there will therefore remain 
only {(n + 1) — (ti — 1)} or 2 coefficients really arbitrary in the terms of the 
nth degree in F. 

Now the expressions 

and F=(ir — iy)* 

satisfy equation (1), and therefore if A^ and B^ ate any arbitrary constants, 

the expression 

An {x + lyY ^Bn{x- iyY 

satisfies equation (1), and is homogeneous of the nth degree in x and y, and 
contains two arbitrary constants. It therefore represents the most general 
form of the terms of the nth degree in V\ and so the general solution of 
equation (1), regular at the origin, can be expressed in the form 

F(a:,y) = ilo + ^(a? + ty) + 5i(a?-iy) + il,(^ + iy)« + A(«-iy)"+ (2), 

where the quantities Aq, A^, B^, A^, ... are arbitrary constants. 

This expansion furnishes the general solution of equation (1) ; what is 
however in general needed is the particular solution of equation (1) which 
satisfies some further conditions. As an example of the conditions most 
fi'equently occurring, we shall suppose that the value of the required solution 
V(x, y) is known at every point of the circumference of a circle, whose centre 
is at the origin and whose radius is any quantity a ; it being supposed that 
this circle lies wholly within the region for which V is regular. This being 
given, we shall shew that the constants Aq, Ai, B^, ... can be found, and 
the solution can be completely determined. 

For writing 

x = r cos 0y y^r sin 0, 

the value of V is known when r = a, as a function of 0, say f(0). Let the 
function /(^) be expanded as a Fourier series in the form 

f(0) = Oo + Oi cos ^-h 6i8in ^ + Oacos 2^ + 63 sin 20 + (3), 

where the coefficients a©, Oi, 61, a,, ... are given by the formulae 



1 r*' \ 




•2ir 



1 [^ 

an = - I /(t) COS ntdt 

IT J Q 

1 f^' 
6n = — / f(t) sin ntdt 



y (*). 



168] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 311 

Consider now the expression 

ao + ^(oi COS ^ + 61 sin 0) + Q (a, cos 2^ + b^ sin 20) + (5). 

This expression (0) reduces to (3), i.e. to /(^), when r=^a\ and since we 
have 

r^cosn0=^ {(x + iy)* + (a? - iy)»}, 

r« sin w^ = 2^ {(a? + iy)« - (a? - iy)**}, 

it is clear that the expression (5) is of the form (2), i.e. that it is a solution 
of the equation (1). 

It follows that the solution V of equation (1), which is characterised 
by the condition that it has the value V=f(0) when r = a, is given by the 
expansion 



where 



r fr\^ 

F= Oo + -(aicos ^ -h 61 sin ^) 4- -) (o^ cos 20 + 6, sin 2^) + ..., 
a \af 



I 1 f^' 



< 



1 f*' 
a% = - I f(t) cos ntdt, 

1 f^ 
bn = - I f{t) sin ntdt 
\ ^ Jo 

The principal object of this chapter will be to obtain theorems analogous 
to this for the other partial differential equations of mathematical physics ; 
the method followed will be in most respects similar to that by which this 
result has been obtained. 

168. Laplace's equation; the general solution; certain particular solutions. 

The partial differential equation 

a»F d^V d'V ^ 
da^ dy^ dz^ 

is known as Laplace's equation, or the potential-equation, and is of importance 
in the investigations of mathematical physics. 

The general solution of this equation was given by the author in 1902. 
It may be written 

r2ir 

F= I f(x cos t-^-y sin t + iz, t) dt, 
Jo 



312 TRANSCENDENTAL FUNCTIONS. [CHAP. Xni. 

where / is any arbitrary function of the two arguments x cos t + y sin t + iz 
and t The solution is eflFected in Monthly Notices of the Royal Astron. Soc, 
VoL LXii. In this chapter however we are -concerned not so much with the 
general solution as with the particular solutions which satisfy certain further 
conditions. To the consideration of these we shall now proceed. 

Let the equation be transformed by taking instead of the independent 
variables a?, y, z, a new set of independent variables r, 0, ^, connected with 
them by the relations 

x^rsmO cos ^, 

y = r sin ^ sin ^, 

z = r cos 0. 

It is found without difficulty* that Laplace's equation becomes 

drV dr)^ siu^e a<^« "^sin 0d0\^ dO)"^' 

Let us seek for particular solutions of this equation, of the form 

V^Re<P, 

where R, 0, <I>, are functions respectively of r alone, alone, and ^ alone. 

Substituting, we obtain 

1 d f.dR\ 1 d ( , ^de\ 1 d^ 

Rdr\ dr)^(bHm0dd T^ ^ dff) "*" cDsm*^ d<^« ""' 

Now the quantity 

Rdr\ drj 

does not involve ^ or ^ ; and since by this equation it is equal to 

1 d / . ^d0\ 1 d^ 

@8in0d0V^ dd) 4>sin^^d<^«* 

it clearly cannot vary with r : it is therefore independent of ?•, ^, and <^, and 
so must be a constant ; this constant we shall write in the form n{n'ir 1). 



^,(^f)-»(.-i)^-o. 



We thus have 

dr 

Write r ^ 6**, so dr = e^du. Then this equation becomes 

d / ., dR^ 



^U''£)-^<^^'^''=' 



d^R dR , , 1 \ z> A 

or -FT +-3 n(n+l)R = 0. 

du^ du ^ 

* The ^ork is given in full in Edwards' Differential Calctdut, 



168] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAI. PHYSICS. 313 

This is a linear diflferential equation of the second order with constant 
coefficients ; its solution, found in the usual way, is 

where A and B are arbitrary constants. 

The most general form of the function R is therefore 

Considering next the function <I>, it can in the same way be shewn that 
the quantity 

is independent of r, 0, and (f>, and so must be a constant. Writing 
this constant in the form — m', we have for the determination of <I> the 
equation 

~— 4- m*4> = 

of which the general solution is 

<I> = a cos m<f> + h sin m<^, 

where a and h are arbitrary constants. 

It thus appears that the expressions 

r"cosm<^0 and r*sin77i^© 

are particular solutions of Laplace's equation, if n and m are any constants 
and is a function (of 6 only) which satisfies the equation 

^ ^ %%\n0dd\ ddj sin*^ 



Writing cos^ = 2r, this becomes 






© = 0. 



But when m is a positive integer, this is (§ 129) the equation which is 
satisfied by the associated Legendre functions of order n and degree m ; so a 
particular solution is the function 

Pn'^iz), or P„«»(cos^). 

Hence generally we see that the (2n+ 1) expressions 
r'^Pn (cos ^), r'* cos <^ Pn^cos ^), r~ cos 20 Pn' (cos ^), ..., r'^C0S7K/>Pn'*(C0S^), 

r" sin <^ Pn"^ (cos 6), r^ sin 20 Pn^ (cos 0), ..., r^ sin iKf> Pn^ (cos 0), 

where n is a positive integer, are particular solutions of Laplace's equation. 



314 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII. 

Moreover, since PfJ^ (cos 0) is of the form sin"* ^ x a polynomial of degree 
(n — m) in cos 0, it is easily seen that each of these quantities, if expressed in 
terms of x, y, z, becomes a polynomial, homogeneous of degree n, in a?, y, z. 
It can in fact be easily shewn, by using the result of § 132, that 

r" cos m<f> Pn^ (cos 0) 
is a constant multiple of 



f 

J Q 



•2ip 

(a? cos t + y sin. t-h iz)^ cos mt dt, 



and that 

r^ sin m<l> Pn^ (cos 0) 

is a constant multiple of 



/, 



Sir 

(x cos t + y sin t-^ izY sin mt dt, 





from which their polynomial character is evident; these forms have the 
further advantage of exhibiting these particular solutions as cases of the 
general solution given at the beginning of this article. 

Example, If coordinates r, d, ^ are defined by the equations 

y = (r* - 1)* sin S cos (f), 

.2= (r*- 1)* sin $ sin <^, 
shew that the function 

7= P^'» (r) P„»» (oos $) cos mit> 

is a solution of Laplace's equation 

169. jf^c series-solution of Laplace* s equation. 

The particular solutions of Laplace's equation, which have been found in 
the preceding article, enable us to express the general solution, in the form of 
an infinite series involving Legendre functions. This series-solution will of 
course be really equivalent to an expansion of the general solution 



/, 



Jir 

fix COS ^ H- y sin ^ + u, t) dt 





already mentioned ; but the series-form is (as will appear from § 170) more 
convenient in determining solutions which satisfy given boundary-conditions. 

For let V(Xj y, z) be any solution of Laplace's equation 

da^ "^ dy^ "*■ dz^ " 



170] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 315 

Then in the neighbourhood of any ordinary point, which we may take as 
the origin of coordinates, V can be expanded in the form 

Substituting this expansion in Laplace's equation, and equating to zero 
the coeflScients of the various powers of x, y, z, we obtain an infinite number 
of linear relations between the coeflScients a©, Oi, 61, Ci, Oj, .... 

There are ^n{n'-\) relations of this kind between the ^(71+ l)(n-h2) 

coeflScients of terms of degi'ee n in the expansion of V: and so only 

-{2 (n + 1) (n + 2) — s w (n — 1)1 or (2n + 1) of the coeflScients of terms of 

degree n in the expansion of V are really independent. But in the last 
article we have found (2n + 1) independent polynomials of degree n in a?, y, z, 
which satisfy Laplace's equation, namely the quantities 

r^Pn (cos 6\ r~ cos <\>Pf^ (cos 0), , r~ cos n<f>Pn^ (cos 0), 

r^Bin (f>Pn^(cm0)y , r^ sin TK^Pn** (cos ^). 

It follows that the terms which are of degree n in a?, y, z in the expansion* 
of V must be a linear combination of these (2n + 1) quantities ; that is, 
V must be expansible in the form 

F = ilo + r [A^P^ (cos 0) + ^1^ cos ^P,^ (cos 0) + B,^ sin <I>P^^ (cos 0)] 

+ r« {A^P^ (cos 0) + Ai^ cos <^P,» (cos 0) + A^* cos 20 P,' (cos 0) 

+ Ba^ sin i^Pa^ (cos ^) + J8,« sin 20 P,^ (cos ^)} + . . . , 

where the quantities A^, A^, A-^y B^y ... are arbitrary constants. 

170. Determination of a solution of Laplace's equation which satisfies 
given boundary conditions. 

In order to determine the unknown constants Aq, Ai^ A^y B^y ..., which 
appear in the expansion just found, it is necessary to know the remaining 
conditions which the function V is required to satisfy. A condition of 
frequent occurrence is that V is to have certain assigned values at the points 
of the surface of a sphere, which we may take as being of radius a and having 
its centre at the origin. This sphere will be supposed to lie entirely within 
the region for which F is a regular function of its argtiments a?, y, z. When 
r = ay F is therefore to be equal to a given function f{0y <f>) of and 0. 
The constants Aq, Aiy -4l^ JS/, ..., are therefore to be determined fix)m the 
equation 

f{0y ^)^Ao + a {ill Pi (cos 0) + ili» cos (f> Pi^ (cos 0) + A' sin <l>Pi^ (cos 0)] 

+ a* {A^Pi (cos 0) + -4,^ cos (fyP^^ (cos 0)+ ...] + ... 



316 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII. 

In order to obtain the value of one of these constants, say -4n'*, from this 
equation, we multiply both sides of the equation by Pn** (cos ^) cos m^, and 
integrate over the surface of the sphere. On the left-hand side we thus have 

r i '/(^» ^) -Pn*" (cos d) COS nKf> sin dd d(f>. 
Jo Jo 

As to the right-hand side, we know that 



/, 



2ip 





is zero except when r = m, and that 

/•2» 



COS m<f> cos r<f> d<l> 



I COS m<f> sin r<f> dif> 

Jo 

is always zero ; and also (by § 130) that 

f ' Pr"^ (cos 6) Pn"" (cos 0) siu d0 
Jo 

is zero except when r = n. It follows that on the right-hand side, every 
term vanishes except the term 

a'^^n"* re {Pn"* (cos 0)Y cos» m</> sin d0 d<f>. 
Jo Jo 

Since I cos' m^ d<f> = tt, 

and (by § 1 30) /J {P„» (cos 0)}^ sin 6 d0 = ^^^ <^| . 

this term has the value 

2n-|- 1 (n — m)! 
We have therefore the formula 

^""^ = i^»' • (^! /o' ly^^- '^^ ^-^ <^°« ^> ""^^ "^<^ «^ ^ '^^ '^*' 

which determines the coeflScients -4n"* in the expansion of V, 

The coeflScients £„"* can be similarly determined : and so finally the 
solution V of Laplace's eqxuition, which has the value f{0, <f>) at the surfa^ of 
the sphere, is given for points in the interior of the sphere by the expansion 

^ = Jo ^^ ^ (a)7«' ly^^'- *^'^ {^" ^"^^ ^^ ^" ^*'*'' ^^ 

+ 2 2 J^^^^;P„'»(co8 5')-Pn"(cosd)cosm(^-^')|8in^d^d<^'. 



171] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 317 

This result may be regarded as a three-dimensional analogue of the two- 
dimensional result of § 167. 

Elxample 1. Shew, by applying the expansion-theorem just given, that 
P^ {cos ^ COS ^ + sin ^ sin ^ COB (<^ - <t/)} — P„ (cos 6) P„ (cos ff) 

+ 2 J^ |J^; P,- (cos e) P,« (cos ff) COS m (* - <t>'). 

Example 2. Prove that if the product of a homogeneous polynomial of d^ree n in 
0?, y, z and the function P„{cos^cos^+sin^sin^cos(^-^')} be integrated over the 
smface of the sphere, the result is 4ir/(2n+l) multiplied by the value of the polynomial 
at the point (^, ^'). 

(This can be proved by taking ^ to be zero, which involves no real loss of generality, 
and expanding the polynomial by the theorem of this article.) 

171. Particular solutions of Laplace's equation which depend on Bessel 
functions. 

It is possible to construct solutions of Laplace's equations in series in 
several ways, of which that which has been given, and which depends on 
Legendre functions, may be taken as representative. A fall discussion of 
the other methods would be beyond the scope of this book, but a general 
idea of them may be inferred from the result which will next be established, 
namely that the Bessel functions furnish a group of particular solutions of 
Laplace's equation, just as the Legendre functions do. 

When Laplace's equation 

da^ dy^ dz* 

is expressed in terms of the " cylindrical coordinates *' z, p, ^, where p and ^ 
are defined by the equations 

\x^ p cos <^, 

[y = p sin ^, 
it takes the form 

dz^ dp" '^ pdp "^ p^d<l>^ 
Let us seek for particular solutions of this equation, of the form 

F = ZP<I), 
where Z, P, <I>, are functions of z alone, p alone, and ^ alone, respectively. 

On substituting this value of F, Laplace's equation becomes 

ld«Z l/d|P ldP\ 1 d«<l> 
Zdz*"^ ?\dp''^ pdp)'^ p^d<f>^'' 



I 



318 TRANSCENDENTAL FUNCTIONS. [CHAP. Xin. 

This equation shews that the quantity 

Z dz^ 

must be a constant independent of z, p, and ^ ; let this constant be denoted 
by i^. Then on solving the equation 

we have the particular solutions 

Z = e** and Z—e"^. 

Similarly the quantity 

1 d^ 
<t>dif>^ 

is a constant, which may be denoted by — m' ; on solving the equation 

we obtain the particular solutions 

<I> = cosm<^ and <I> = 8inm^. 
The equation to determine P is now 



>-Jfn'- $)--«■ 



dp 
On putting kp = y, this becomes BesseUs equation of order m, 



*''+i^+ii 



-?")'-»■ 



df y dy 
a particular solution of which is 

P = /m(y). 

It follows that the expressions 

e"^ cos 7n<f>Jm{lcp) and ^±** sin m^t/in (A;p), 

where k and m are arbitrary constants, are particular solutions of LapUice's 
equation, 

172. Solution of the equation 

— + ~ + 7=0. 

We now proceed to consider another partial differential equation. 



173] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 319 

We have seen in the last article that Laplace's equation 

da^ dy'* dz* 
is satisfied by the particular solutions 

^Jn(r) cos 710 and e* /„ (r) sin w^, 
where a? = rcos^, y = r8in^. 

But if we write TT^eT, 

where F is a function of x and y only, the Laplace's equation for W becomes 

It follow^ that, for all values of n, the quantities 

Jn (r) cos nO and /» (^) sin nO 

are particular solutions of this latter equation.. 

From these particular solutions, as in the case of the solution of the 
equation 

already described, we can build up the general solution of the equation 

d^V 3'F 

— + 4- F=0 



00 



in the form V= ^ J^ (r) {an cos nO + bn sin nO), 



n=»o 



where ao, Oi, a,, ..., 6i, b^, ..., are arbitrary constants. 

173. Solution of the equation 

3»F d*V a»F ^ ^ 
1 4- — I- F = 

In order to solve the equation 

^ a«F 3«F 

which is likewise of great importance in the investigations of mathematical 
physics, we first express the equation in terms of new independent vaiiables 
r, 0, (f>, defined by the equations 

a; = r sin ^ cos (f>, 

y=^r sin sin (f>, 

,z =rcos^, 



320 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII. 

and theD endeavour to find particular solutions of the form 

F=iee<i>, 

where -B, ©, <I>, are functions respectively of r alone, 6 alone, and <^ alone. 
Proceeding as in § 168, the diflFerential equation becomes 

1 d/ d^N 1 d ( . ad^\ . 1 d^ 

■*■ 22 rfr V dr j "^ sin ^ d^ V®*" df>; '•' 4> ain» ^ d^» ° "• 

This equation can be solved by the process used in § 168 for finding 
particular solutions of Laplace's equation ; the quantity 



^ RdrK dr) 



must be a constant, which we shall denote by n (n + 1). If in the resulting 
«quation 



we write y = Rr^, it becomes 



(. . !)1 



which is BesseUs equation of order (^ + s) • 

The quantity R can therefore be taken to be 

R^r^Jn^^{r), 

The equations for % and <I> are now found to be the same as those which 
occur (§ 168) in the solution of Laplace's equation; and proceeding as in 
§ 169, we find that the general solution of the partial differential equation 

^ 9>F a»j 

regular near the origin, can be expressed in Reform 
F= i r-iJn^{r) 



n=o 



AnPn (cos 0) -h An' COS <t>Pn' (cOS ^) + . . . + An"" COS n<l>Pn'' (cOS 0)] 

-I- Pn' sin <f>Pn' (cos ^) + . . . + 5„« siu TK^Pn** (cOS 0) ) 



+ Bn' si] 
where the quantities A and B are arbitrary constants. 



173] APPUCATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 321 

When a particular solution V of the equation is to be determined by the 
condition that it is to take prescribed values at all points on the surface of a 
sphere, the constants A and B are determined exactly as in § 170. 

Example, Shew, as a case of the general expansion of this article, that 



»sO 



Note, The partial differential equations of §§ 172, 173, possess general solutions 
analogous to that of Laplace's equation. The solution of the equation of § 172 is 

where /is an arbitrary function ; and the solution of the equation of § 173 is 

where / is an arbitrary function. For the proof of these results, reference may be made 
to papers by the author. 

Miscellaneous Examples. 

1. If a solution V of Laplace's equation be symmetrical with respect to the axis of i, 
and have the value V^f(z) at points on that axis, shew that its value at any other point 
of space is 

F-i f '/{«+» (^+y')* cos*} ^• 

tr Jo 

2. Deduce from the result of Example 1 that the potential of a circular ring of 
mass M, whose equation is 



IS 



- f'if[c«+{«+i(a:»+y*)*co8<^}«]"*(i0. 

^ J 



3. Let P (x, y, ;r) be a point in space, and let the plane through P and the axis of z 
make an angle ^ with the plane ix. Let this plane cut the circle whose equations are 

in the points a and y, and let the angle aPy be denoted by $ and log (Pa/Py) by <r. 

If <rj Bf 4>'be regarded as coordinates defining the position of the point P, shew that 
Laplace's equation 

^ a«F a«F 

takes the form 

d_ ( 8inh«r ?n . 1 [ sinh a 8 Fl 1 ^^o 

9<r toosh«r-coe^ 8<rJ 8tf lco8h<r-cofl^ ^j sinh^ a (cosh a - cos $) d<f)^ ~ ' 

and that the quantities 

F— (cosh o--coa tf)* cos «^ cos WM^i^_, (cosh c) 
are solutions of it. 

W. A. 21 



CHAPTER XIV. 

The Elliptic Function ff (z), 

174 Introduction. 

li f{z) denote any one of the circular functions sin z^ cobz, tau ^ ... , it is 

well known that 

f(z + 2,r) =/(^), 
and hence that 

f{z + 2n7r) =/(^), 

where n is any positive or negative integer. 

This fact is generally expressed by the statement that the circtdar 
functions admit the period 2ir, They are on this account said to be periodic 
functions; and in contradistinction to other classes of periodic functions, 
which will be introduced subsequently, they are called singly-periodic 
functions. 

It will in fact be established in this chapter that a class of functions 
exists possessing the following properties : if f(z) be any function of the 
class, then f{z) is a one-valued function of z, with no singularities other than 
poles in the finite part of the ir-plane ; moreover, /(-?) satisfies, for all values 
of z, the equations 

/(* + 2a,.) =/(«), 

where cdi and ©j are two quantities independent of z. Functions f{z) of 
this class are said to admit the quantities 2(!i)i and 2o>2 as periods, and are 
called doubly-periodic Junctions or elliptic functions. The two periods 2(»i 
and 2a>s pl^y ^^^ same part in the theory of elliptic functions as is played 
by the single period 27r in the theory of circular functions. 

By repeated application of the formulae written above, we obtain as the 
characteristic equation of all elliptic functions the equation 

f{z + 2mft), + 2no)8) = / W» 

where m and n are any integers. 



174, 175] THE ELLIPTIC FUNCTION ^{z). 323 

176. Definition of |> {z). 

The elliptic functions may^ as we have just seen, be regarded as a 
generalisation of the circular functions. It is natural therefore to introduce 
them into analysis by some definition analogous to one of the definitions 
used in the theory of circular functions. 

One mode of developing the theory of the circular functions is to start 

from the infinite series 

1 ** 1 

z^ m^±i{z-miry' 

It can be shewn that this series converges absolutely and uniformly for 
air values of z except the values 

-8^ = 0, ±7r, ±27r, ±37r...; 

And that it admits the period 27r. If now its sum be denoted by (sin ^)~*, 
and this be regarded as the definition of the function sin z^ then from this 
definition we can derive all the properties of the function sin z^ and thus 
a complete theory of the circular functions can be developed. 

Similarly, as the basis of the theory of elliptic functions, we form the 

infinite series 

r^ + 2 {(^ - 2ma)i - ^nto^-^ - (2mft), -h 2n<»a)-«}, 

^here a*i and o), are any two quantities, independent of z, whose ratio is not 
purely real, and where the summation extends over all integer and zero 
{except simultaneous zero) values of m and of n. 

It has been shewn in § 11 that this series is absolutely convergent for all 
values of z, except the values -^ = 0, ± Wi, ± Wj, + o), ± o),, ± 2©! ± ©a, ... . 

By comparing the series with the convergent series 2(tn* + n')~* as in 
•§11, it is seen that this convergence is also uniform (§52). The series 
therefore represent'S a one-valued function of z, regular for all values of the 
variable z except the values z = ^mo)^ + 2no>, ; and at these points, which are 
the singularities of the function, it clearly has poles of the second order. 

We shall denote this function by the symbol fp^z). Its introduction is due 
to Weierstrass. 

There are other ways of introducing both the circular and elliptic fiinctions into 
Analysis ; for the circular functions, the following may be mentioned : 

(1) The geometrical definition, according to which sin z is the ratio of one side to the 
hypotenuse, in a right-angled triangle of which one angle is z. This is the definition usually 
given in the introductory chapter of treatises on Trigonometry : but from our point of 
view it is defective, as it applies only to real values of z. 

(2) The definition by means of the infinite product 

sin«-z(l-5)(l-2gi)(l-^).... 

21—2 



324 TRANSCBNDBNTAL FUNCTIONS. [CHAP. XIV. 

(3) The definition by the inversion of a definite integral, 

We shall see subsequently that alternative definitions of the elliptic functions exist, 
analogous to each of these definitions (1), (2), (3), and that they may if desired be taken 
as fundamental in the theory. 

Example, Prove that 

••<"-<'K£.)'.L~~^('-^')' 

176. Periodicity t and other properties^ of |> {z). 

The function ^(z) is an even function of z^ Le. it satisfies the equation 

For if —z be substituted for z in the series which defines |> {z\ the 
resulting series is the same as the original series, except that the order of 
the terms is changed. But since the series is absolutely convergent, this 
change in order does not affect the value of the sum of the series; and 
therefore we have 

if>(^)=if>(-'^)- 

Further, the function |f> {z) admits the quantity 2o>i as a period. 
For 

= (2: + 2(k)i)-» - r-« 4- S {(^r + 2a)i - 2ma)i - 2no>^y^ - (^ - 2mo)i - 2n<»a)"'} 

= S {(2r - 2 (m - 1 ) Oh - 2na)8)-« -{z- ^nuo^ - ^am^)-^] . 

where the last summation is extended over all integer and zero values of m. 
and n without exception. But this last sum is zero, since its terms destroy 
each other in pairs. Thus we have 

Similarly f{z+ 2(k)a) = ^ {z\ 

and generally V{^ + imcoi -h 2rw«>g) = fp (z), 

where m and n are any integers. 

Therefore the function fp(z) admits the two periods 2ft)i and 2w2. 

Differentiating the above results, we see that jf>' (z) is an odd function of 
z, and admits the sams periods as ^ (z). 

177. The period'paraUelograms, 

The study of elliptic functions is much facilitated by a method of 
geometrical representation which will now be explained. 



V^PP 



176 — 178] THE ELUPnc function ^ (z). 325 

Suppose that in the plane of the variable z we mark the points z^O, 
z s= 2(i(>x, z a 2(!i)9, z = 2a>i + 2a>s, ... and generally all the points comprised in 
the formula z = ^ramx + 2na>8, where m and n are any positive or negative 
integers or zero. 

By joining the point ^ = by a straight line to the point ^ = 2o>i, then 
joining the point 2o)i to the point 2(k>i + 2o)„ then joining the point 2o)i + 2(0, 
to the point 2a>s, and lastly joining the point 2a>9 to the point ^ = 0, we 
obtain a parallelogram in the ir-plane, which we shall call the fundamental 
period-parallelogram. 

It is clear that the whole ^-plane may be covered with a network of 
parallelograms, which are each similar and equal to this parallelogram, and 
which can be obtained by joining the other marked points by straight lines. 
These parallelograms will be called period-parallelograms. 

Then if < be any quantity, the points 

z^t, ^ = t+2a>i, 5«5t+2a>2, ..., £: =» ^ -h 2tna)i + 27Mi),, 

manifestly occupy corresponding positions in these parallelograms; these 
points are said to be coryruent to each other. 

It follows from the fundamental property of ^{z) that the fanctum f(z) 
has the same valvs at all points which are congruent unth each other ; and 
hence that the values which the function fp{z) has in any periodrparallelogram 
are a mere repetition of the values which the function has in any other period- 
parallelogram. 

178. Expression of the function fp (z) by mea/ns of an integral. 

We shall now obtain a form for |> {z) in terms of an integral, which will 
be found to be of great importance in the theory of the function. 

The quantity (p {z) - r^, 

or 2 [{z — 2nK0i — 2na),)"* — (2mah -h 27k»,)~*}, 

is a regular fuuction of z in the neighbourhood of the point z^O, and is an 
even function of z. It can therefore by Taylor's theorem be expanded, for 
points z near the origin, in the form 

where clearly we shall have 

^ = 32 (2mo>i -h 2no)a)-^, 

^ = 52 (2mo>i + 2wo>a)^. 

Thus j>(^) = r-+g^ + g^ + .... 



826 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV. 

Forming the square and the derivates of this expansion, we have 

|»"(«)-6*-^ + ^ + |^,*»+.... 

Therefore jf** («) — s if>"(^) " n^'* **" **™"8 involving z* at least. 
It follows that the function 

is regular in the neighbourhood of the point z^0\ and as it is doubly-periodic 
(for clearly any power or derivate of an elliptic function is likewise an elliptic 
function) it must be regular in the neighbourhood of each of the points 

z «s 2mcDi + 2n(k>s. 

But the only singularities of ^{z) are at these points : and therefore the only 
possible singularities of the function 

are at these points. The latter function is consequently regular for all values 
of z\ and so by Liouville's theorem (§ 47) is independent of z, and therefore 

is equal to the value which it has at the point z^O, which is j^^s* 
We have therefore the relation 

P'(^)=^ii»"(^) + r2^.. 

Multiplying by 3 jf>' {z) and integrating, we have 

where c is a constant ; on substituting the expansions in this equality, we 
find that c = ^g^. 

Thus, finally, the function p (z) satisfies the differential equation 

jp'» (^) = 4 jf>> (^) - 5r, jp (^) - 5r„ 

where g^ and g^ (called the invariants) are given in terms of the periods 
of jp(<j) by the equations 

^r, = 60 2 (2ma)i + 2no),)~*, 

5r, = 1402 (2?na)i + 2na),)"*. 



178] THE ELLIPTIC FUNCTION fp(z). 327 

This differential equation can be written in the form 

where ^ »= j? (^)i 

and therefore (since ^ (z) is infinite when z is zero) we have 

which is the required expression of ^(z) in terras of an integral. 

The preceding theorems may be illustrated by the results which correspond to them in 
the theory of the circular functions. Thus we may in the following way discuss the 
properties of a function f{z) (really ca$ec^z)f which we shall take to be defined by the 
series 

This series is clearly infinite at the points ^^O, ir, - ir, 2ir, ... ; for other values of z it 
is absolutely and uniformly convergent, as is seen by comparing it with the series 

H-l-«+l-«+2-a+2-«+3-«+3-*+.... 

The effect of adding any multiple of ir to ^ is to produce a new series whose terms are 
the terms of the original series, arranged in a difierent order ; this does not affect the sum 
of the series, since the convergence is absolute ; and therefore/ (^) is a periodic function of 
X, with the period w. 

By drawing parallel lines in the 2-plane at distances w from each other, we therefore 
divide the plane into strips, such that at points occupying corresponding positions in the 
different strips, /(f) has the same value. In each strip, f{z) has only one singularity, 
namely at that one of the points 0, ir, — «-, 29r, -29r, ... which lies within the strip. The 
function is not infinite at the infinite ends of the strip, because the several terms of the 
series for f{z) are then small compared with the corresponding terms of the comparison- 
senefi 

1 +l-«+l-«+2-«+2-a+3-«+3-«+... . 

Now near the point f =0, the function /(«) can be written in the form 

/«='-*+'-'(l-i)"*+'-'(l+^)'*+(2')-'(l-2^) "*+... 

=*-*+ir-*(l+l + 2-*+8-«+...)+»r-M(3+3+3. 2-« + 3. 2-«+. ..)+.. . 
=«-»+2«-« . ^+,r-**». 3 . 2 . ^+... 

Differentiating and squaring this equation, we have 

It follows that 

/"W-6/«(*)+4/(*) 



328 1*RANSCENDENTAL PUNCTIONa [CHAP. XIV. 

is a series containing no negative powers of z; it has therefore no singularity at the point 
^^■■0, and therefore (since that is the only possible singularity) no singularity in the strip 
which contains 2=0, and therefore (on account of the periodic property) no singularity in 
any strip. It is therefore, by Liouville's theorem (§ 47), a constant : this constant must 
be equal to the value of the function at the point ^--O, which (on substituting the expan- 
sions) is found to be zero. We have therefore 

/"W-6/«W+4/(,)-0. 
Multiplying by 2/' (t) and integrating, we have 

where c is a constant. On substituting the expansions, c is found to be zero, and therefore 

/'»«=4/«(i){/(.)-l} 

or (£)*-^ (^- ^)' ^^^ ^-/W> 

which gives 2«- / f^{t- 1)"* dt 

as the expreesioD of f(s) by means of an integral 
Example, If y = ^ («), shew that 

\dz) \ck) 

where «i, «s, «} are the roots of the equation 

For we have f(^('^)'^«^(')-9fi?(s)^9zf 

and so (^y-4(y-«i)(y-«,)(y-6j. 

Differentiating logarithmically, we have 

''^-(y-«i)-^+(y-«j)-^+(y-«s)-^. 



Dififorentiating again, we have 



ds^ \dz*) 



\dz) \aSJ 

Adding the last equation, multiplied by ^, to the 49quare of the preceding equation, 
multiplied by fj, we have the required result. 

It may be noted that the left-hand side of the equation is half the Schwartzian derivative 
of z with respect to y ; and hence the result shews that z is the quotient of two solutions 
of the equation 



179, 180] THE ELLIPTIC FUNCTION ff (z). 329 

179. ITie homogeneity of the function fp (z). 

When the Weierstrassian elliptic function is considered as depending on 
its arguments and periods, it has a certain property of homogeneity, which 
will now be investigated. 

Let fp Iz, M denote the function formed with the argument z and periods 
2a>i and 2^2. Then we have 

It follows that the effect of multiplying the argument and the periods by 
the earns quantity \ is equivalent to multiplying the function by X"^. 

This relation can also be expressed in terms of the quantities gtt g^ 

For let fp(z; g^, g^) denote the function formed with the invariants 
g^ and g^. Then we have 

g^ss 60 2 (2mo)i + 2w<tt,)"*, 

^fj =s 1402 (2mft>i + 2nG)a)~*. 

The effect of replacing a>i and o>s by Xq>i and Xto^ respectively is therefore 
to replace g^ and g^ by X^g^ and X~*5r, respectively; and thus we have 



K';s„}.)-f{'. ^) 



M"- ^ 



= X»jf>(X(r; X-*g„ \-*gt), 

m 

which expresses the homogeneity-property in terms of the invariants. 
Example. Deduce the last result directly from the equation 



180. The addition-theorem for the function fp (z). 

The function fp(z) possesses an addition-tiieorem, i.e. a formula which 
gives the value of |> (xr + y) in terms of the values of fp (z) and fp (y), where 
JB and y are any quantities. 



4f 

•4 



830 



TRANSCENDENTAL FUNCTIONS. 



[chap. XIV. 



To obtain this formula, consider the expression 

,1 F(*) 9'^') 

! 1 «»(y) F'(y) 

as a function of z. 

Since it is compounded of doubly-periodic functions, it is itself a donbly- 
periodic function ; and the only points at which it can have singularities are 
the points at which the functions jp(^ + y) and ^{z) have singularities, 
i.e. the points ^ = 0, z^^y, and points congruent (§ 177) with these. 

Now for points z near the point ^ = 0, we can write the determinant in 
the form 

1 ip(y) + ^|)'(y) + W'(y) + ... -i?'(y)-^j?"(y)-... 



^+Mfl^«^+ — 



-2r-»+j^flra« + .., 



1 viy) ip'(y) 

Expanding this determinant, we find that the terms involving negative 
powers of z destroy each other; the determinant can therefore, in the 
neighbourhood of the point ^ = 0, be expanded as a series of positive powers 
of z ; that is, the function represented by the determinant has no singularity 
at the point z^O; and therefore (by the periodic property) it has no 
singularity at any of the points congruent with z^O. 

Considering next the neighbourhood of the point z^ — y, write ^ = — y + a?. 
The determinant can be written in the form 

1 jf>(y) «>'(y) 

and on expansion this is found to contain no negative powers of x. The 
function represented by the determinant has therefore no singularity at the 
point xr = — y or any of the congruent points. 

The function has therefore no singularities, and so by Liouville's theorem 
(§ 47) is independent of z. But it vanishes when z has the value y, since two 
rows of the determinant are then identical.. The determinant is therefore 
always zero. 

We thus have the formula 



1 ^{z + y) -p'(xr + y) 

1 «>(y) v'iy) 



= 0, 



180] 



THE ELLIPTIC FUNCTION ^{z). 



331 



true for all valnes of z and y. Since, by § 178, jf^ (^ + y)> §>' {^\ j?'(y) ^re at 
once expressible in terms of jp(-f + y), jp(^), |>(y), respectively, this result 
really expresses ^{z-k-y) in terms of ^{z) and jf>(y). It is therefore an 
addition-theorem. 

The addition-theorem may also be obtained in the following way. 

Take rectangular axes Oxy Ou, in a plane ; and consider the intersections of the cubic 
curve 

with a straight line 

The abscissae Xi, x,, x, of the points of intersection are the roots of the equation 

<^(x)=0, where 

<p (x) « {mx + n)« - 4c5 +^^ +^,. 

The variation liXr in one of these abscissae, consequent on small changes dm and dn in 
m and n, is therefore given by the equation 



<f/ (Xr) 4x^+2 (mxr + n) (Xi^w + dn) ■■ 0, 



whence 



» to. 



ft i x-dm + d» 
— 2 2 



Therefore 



bO, by a well-known theorem in partial fractiona 

8 



Now when n is infinite, the abscissae x^, x^, x^ are all infinite : we may therefore 
int^;rate the last equation over the series of positions of the straight line ytamx-^-ny and 
obtain the result 



3 /•• 

2 I {AXr^^-g^r-g^'^dXr^O. 

r-1 y aL 



If we write 



*i-j?W, *a=j?(y)» *8-if>(t«^), 
we have therefore z •\-y + tr ■■ 0. 

But the ordinates of the three points of intersection are 

«,-!>'(,), Wj-ip-Cy), «,-if)'(to). 
Since the three points are collinear, we have 

I ^3 W3 -0, 

I jp(y) r(y) 

which is the addition-theorem. 



and therefore 



332 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV. 

181. Another form of the addition-theorenL 

The determinantal form of the addition-theorem given in the last article 
may be replaced in the following way by a simpler, though less symmetrical, 
formula. 

Consider the equation 

1 jp(a.) f(/(w) =0. 

1 j?(y) jf>'(y) 

If in this we replace fp> {x) by its value in terms of |> (x), and expand, 
we have 

{4t>» (w) ^g,fp (x) - g,\ { jp (z) - 1> (y)}« 

= [fp' W {p (^) - j? (y)} + jf>' (y) {» (^) - 1> (^)}?. 

This may be regarded as a cubic equation in the quantity |> (x). One of 
its roots is jp(^) = |>(^+y), by the addition-theorem; and the other two 
roots are jp (^) = jp (z) and fj>{x)^jp (y), since the determinant vanishes when 
x^ or y is substituted for x. We have therefore 

|> (z) + |p(y) + ip(^ + y) = Sum of roots of cubic 

= - {Coefficient of p«(a;)} -r {Coefficient of f^(x)} 

= i {J?' (^) - j?' (y)}M«> W - j? (y)}"*, 

and thus we have 

which is a new form of the addition-theorem. 
Example 1. Prove that the expression 

i{r W - r (y)}' {J? W - f (y)} -* - «» W - f («+y). 

considered as a function of z^ has no singularities: and deduce the addition-theorem 
for ip(z). 

For the given expression, from the mode of its formation, can clearly have no singu- 
larities except at the points e=xO, z=yy z^-y, and points congruent with these. 

Consider then first the neighbourhood of the point z—0. The expression can be 
expanded in the form 

-P(y)-«r(y)--. 

and this on reduction is found to contain no negative powers of e, the first non-zero term 
being fp (y). The expression has therefore no singularity at the point ^=0. 



181, 182] THE ELLIPTIC FUNCTION ^{z). 383 

Considering next the neighbourhood of the point z^y^ we take fny+x ; the ezpressioQ 
becomes 

i {fr' (y)+*r (y)+ - - r (y)}Mf (y)+««>' (y)+ - - P (y)}-'- if (y) -*r (y) - - 

-I»(2y)-xif>'(2y)-..., 

and this on reduction is found to contain no negative powers of x ; there is therefore no 
singularity at the point z^y. 

The case of the point z^ — y can be similarly treated. 

■ 

The given expression has therefore no singularities, and so by Liouville's theorem is 
independent of z. But its value at the point z^O has been shewn to be ^ (y). We have 
therefore, for all values of Zy 

i{P''w-r(y)}*{i»W-jP(y)}-*-«>«-P(*+y)-«'(y)-o, 

which is the addition-theorem. 
Example 2. Shew that 

P(«+y)+«>(«-y)- {if>«- jp(y)} -»({2PW (PW-igt) {|»W+i»(y)}-s'J. 

For by the addition-theorem we have 

Replacing fp^{z) by 4pW-5r,p(z)-^„ and replacing f^{y) by 4jf>8(y)-^,if>(y)-^3, 
and reducing, we obtain the required result. 

182. The roots e^, e,, e^. 

Let n denote any one of the periods of g> {z), namely the quantities 
2a),, 2a)3, 2q>i + 2ft)a> 2a)i — 2g)2> "- 2a)i — 2ca2, .... Then 

§>' (^ fi^ = j>' Q ft - n) , since jf^ (^) has the period ft, 

= — p' ^2 ft j , since |>' is an odd function of z. 

It follows from this that unless 2^ ^^ itself a period (in which case 
jp'T^ftj is infinite), jp'Uftj is zero. 

We have therefore 

|>'(«i) = 0, jf>'(«,) = 0, i?'(a>,) = 0, 

where &>« stands for — (oh + ««>2)- 



334 ' TRANSCENDENTAL FUNCTIONS. [CHAP. XIV. 

Now denote the quantities jf>(a>i), |>(a>s), jf>(G)«) by ^, e,, «,, respectively. 
Then the equation 

jp'» (a)i) = 4|l* (a)i) - 5r, jp (o),) - ^r,, 
or 0=4^»-5r8ei-5r,, 

shews that ^i is a root of the cubic equation 

Similarly e^ and ^s are roots of this equation. 

Moreover, the quantities ei, e,, e, are distinct roots of the equation; for if 
for example we had 61 = ^s, we should have fp (q>i) = fp (q>i), and therefore 

o), = ± 0)1 + a period, 
which is not the case. 

We see therefore that the three roots of the cvhic 

are ei, eg, c,, where 

ei^ip (ft>i), «i = j? (wi), c, = p (o),), 
and ft)i + ft>2 + ®j = 0. 

The quantities ^1, es, ^ therefore satisfy the relations 

ei + «, + es = 0, 
_ 1 

1 

183. Addition of a half period to the argument of ^ (z). 
From the addition-theorem we have 

ip{z + G)0 + ip(2) + 61 =5|>'*(^) {jf>(^) - eij-« 

= {if> (^) - (h] lip (z) - e,} {^ (z) - ex}-S 
or j?(^ + fth) = ^ + («i - «i)(^ - <?i) (ipW - «i}"'. 

This formula expresses the result of adding a half-period to the argument 
of the Weierstrassian elliptic function. 



183, 184] THE ELLIPTIC FUNCTION ^ (z), 335 

Example 1. Shew that 

is a multiple of the dlBcriminaut of the ec[uation 
For we have 

jf> (Z + »i) - «! = (61 - «jj) («! - «3) {P («) - <H}-1. 

Differentiating, we have 
Therefore 

- 16 (ej - e^y (<52 - tfj)* (63-61)2, 
which is a multiple of the discriminant of the equation 

4 (^-61) (^—62) (^-6^=0. 

Example 2. Shew that 
{jP(2^)-«i}{j?(2«)-«J+{«>(2^)-«i}{i?(2*)-6,}+{P(2»)-eJ{p(2^)-63} = |>(«)-^ 

184. Integration of (oa?* + 46^;* + Qca^ + 4(fo? + eY^. 

We shall now shew how certain problems in the Integral Calculus, whose 
solution cannot be found in terms of the elementary functions, can be solved 
by aid of the function |> {z). 

Let the general quartic polynomial be written 

f{x) = our* + Aba^ + 6ca? + ^dx + e. 

Let its invariants* be 

g^sioe — 46d + 3c*, 



9t^ 



a b c 
bed 
c d e 



= ace + 2bcd - c» - (wP - i'^e ; 



let its Hessian be 

A(a?)«(ac - 6«)a?* + 2 (ad - 6c)a;» + (ae + 2bd - S<^)a^ 
-4- 2(6e - cd)a? + (c6 - cP), 
and let its sextic covariant be 



t(^)^l {-/(^) A' (^) + A (^)/' (^)} 



= (a"d - 3a6c + 26«)aJ«H- .... 

* The student who is not already familiar with the elements of the theory of binary forms is 
referred to Bomside and Panton's Theory of Equatiom^ where the invariants and oovariants 
of the quartio are discussed. 



336 TSANSCENDENTAL FUlfCTION& [CHAP. XI7. 

Then it is known that 

<• (^) = - 4A' {x) + g,f* {x) h (x) - (7,/« (x). 

If we write « = — A (x)//{x), this relation becomes 

i^{x)^f*{x)i^-g^-g,). 
Now ^,M.)/-q-y (.)/(.) ^ 

and so (4«' — 5^^ — ^r,)"* cfe = 2 {/(a?)}"* cte. 

Let Xq be any root of the equation /(w) =» ; then to the value a? = o!^ 
corresponds « » 00 ; and hence, if we write 

z=r{f(x)}-idx, 

we have 2z « / (4^ — g^ — fftY^ dt. 

It follows that ths eqimtion 

»{^z\ 9z. 9t)^-h{x)lf{x) 
18 an integrated form of the equation 

= / {cur* + 4iba^ + Qcaf^ + ^dx + e}-*(fo?. 



z 



Example I. Shew that (with the same notation) 

P'(2«;ir,.fl',)-T<(*){/(x)}-i. 

Example 2. Shew also that, if 

then fp (z-i-y) and |>(^~y) are the roots of the equation 

where F(Xy u)—(M^* + 26a««(47+w)+c(:F*+4PM+w*)+2</(4;+v)+e, 

and H {x, u) is derived from h (x) in the same way as F{Xj u) from/(j?). 

(Cambridge Mathematical Tripos, Part II, 1896.) 

186. Another solution of the integration-problem. 

The integration discussed in the last article may also be effected in the 
following way. 

As before, let 

z=r{/(x)}-idx. 

J xo 



185] THE ELUPTIC FUNCTION |> (^). 337 

where f{x) =^00* + Aba? + Qca? + 4dx + 6, 

and let ^0 l>e a root of the equation /(ar) = 0. 
Then, by Taylor's theorem, we have 

f{x) = (^ - x,)f {x,) +\{x- x,yr M + g (^ - a:.yr M 

+ i (^ - ^oYr" M 

Writing (x — Xq)"^ = f» we have 

/ w = i-* {/ ('t-.) ?• + \ f" i^o) ? + 1 r {?=.) r + i /"" (^ . 

and 80 * = /*{/' (^.)(? + 5/"(^,)r' + 6/"'(^.)? + ^/""(^"*d?. 

Writing f = 4 {/'(ar,)}-' ^, we have 

" = /," {4^ + 5 /" (^o) ^ + ^ /' (^•)/"' (^•) ^ + 24 Te/'* C^') /"" (^ ' ^^- 
Now take a new variable of integration «, defined by the equation 

this substitution destroys the term involving the square of the variable of 
integration in the denominator, and we thus have 

where 

5'. = ^/"'(«'.)-h/'(^'>)/"'(«'«)' 

^. = h {/' (^o)/" (^•) /'" (^o) - \ /"• (^o) - 1 /'* (^•)/"" H • 

It can easily be verified that these latter quantities are the same as the 
invariants g^ and g^ of the last article. 

We have therefore 
and therefore ^ "= if^ l-^) "" 24 ^" ^^^* 

?=4{/'(^.))-'jif>W-i^/"(^, 

and finally x = x,^\f' (x.) jjp {z) - ^ /" (x,)} "' . 

This last equation is the integral-equivalent of the equation 

z=r{fix)}-idx. 

w. A. 22 



338 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV. 

It may be observed that 

$>' (z) = (4s» - <7^ - g,)i = i/' (x,) [/ix)]i ?», 
and hence that 

Example, Shew that the integrated form of the equation 

'hdx, 






where ^o ^ ^^7 constant (not necessarily a root of f{x))y and f{x) is any qnartic function 
of X is 

where (f> is the Weierstrassian elliptic function formed with the invariants g^ and g^ 
of/(^). 

Shew further that 

186. Unifomiisation of curves of genua unity. 

The theorem of the last article may be stated somewhat diflTerently 
thus : 

If two variables y and x are connected by an equation of the form 

y« = cwc* + 4iba^ + 6ca?* + 4(ir + e, 

then it is possible to express them in terms of a third variable z by means 
of the equations 

y = i /' M <»' (^) {fr> (^) - r4 /" (^ " ' 

where f{x) = aa^ + ^ba^ + 6ca^ + 4da? + e, 

Xq is any root of th^ equation f{x) = 0, and the function ^(z) is formed with 
the invariants g^ and g^ of the quarticf{x) ; moreover ^ the quantity z is defined^ 
by the equation 

z^\''[f{x)]-^dx. 

Now y is a two- valued function of x, since the quantity 

± (cwr* + 46ar» + 6ca^ + ifdx + e)* 



186] THE ELLIPTIC FUNCTION jf>(^). 339 

may take either sign ; and x is a four- valued function of y, since the equation 

in X 

aa^ H- 46^5* + ^ca? + ^dx + (e - y«) = 

has four roots. But on referring to the equations which express x and y in 

terms of z^ we see that x and y are one-valued functions of z. It is this fact 

which gives importance to the variable z\ zis called the uniformising variable 

of the equation 

y* = oar* + 4iba^ + Qca^ + ^dx + e. 

The student who is acquainted with the theory of algebraic plane curves will be aware 
that curves are classified according to their genvs*^ a number which may be geometrically 
interpreted as the difference between the number of double points possessed by the curve 
and the maximum number of double points which can be possessed by a ciuve of the same 
degree as the given curve. Curves whose genus is zero are called unicursal curves ; if 
f(Xy y)=0 is the equation of a unicursal curve, it is known that x and y can be expressed 
in the form 

» 

where <f> and ^ are rational functions of their argument ; since rational functions are always 
one-valued, it foUows that the variable t thus introduced is the uniformiting variable for 
the equation /(j;, .y)=0 ; i.e., although y is in general a many- valued function of x, and x 
is a many-valued function of y, yet x and y are one-valued functions of z. 

Considering now curves whose genus is not zero, let 

f{x,y)^0 

be a ciurve of genus unity. Then it can be shewn that x and y cau be expressed iu 

the form 

'x=<^) {z) 

where and ^ are now elliptic functions of their argument z ; x and y are thus expressed 
as one- valued functions of 2, and z is the uniformising variable of the equation /(^, y)=0. 
This result is obtained by writing 

where ^and O are rational functions of their arguments, and choosing ^and O in such a 
way that the equation /(:>;, y)KO is transformed into an equation of the form 

we can then wnte 

and X and y will thus be expressed as one- valued functions of z. 
When the genus of the algebraic curve 

n^yy)=o 

is greater than unity, the uniformisation can be effected by means of automorphic 
functions. Two classes of automorphic functions are known by which this uniformisation 

* In French genre^ in German OetchUcht. 

22—2 



c 



340 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV. 

may be effected : namely, one which was first given by Weber in Qdttinger NackriclUen, 
1886, and one which was first given by the author, Phil. Trans., 1898. In the case of 
Weber's functions, the " fundamental polygon " (the analogue of the period-parallelogram) 
is "multiply-connected," i.e. consists of a region containing islands which are to be 
regarded as not belonging to it. In the case of the functions described in Phil. TVatis., 
the fundamental-polygon is "simply-connected," i.e. is the area enclosed by a polygon. 
This latter class of functions may be regarded as the immediate generalisation of elliptic 
functions. 



Miscellaneous Examples. 

1. Shew that 

«>(^+y)-j?(«-y)=-rwr(y){i?W-i?(y)}-*. 

2. Prove that 

where, on the right-hand side, the subject of differentiation is symmetrical in «, y, and w. 

(Cambridge Mathematical Tripos, Part I, 1897.) 

3. Shew that 



r'(^-y) r'(y-^) r'o^-«) 
r(^-y) r(y-^) r(^-^) 

P(^-y) j?(y-tz^) i?(t^-«) 



=i^2 r'(^-y) r'(2/"^) f?"'(w-z) 

iPi'-y) PCy-^^) ip{w-z) 
11 1 

(Trinity College Scholarship Examination, 1898.) 



4. If 



simplify the expression 

where c^, e.^, e^ are the values of jf> (z) for which J>' («)==0. 

(Cambridge Mathematical Tripos, Part I, 1897.) 
5. Prove that 

2{PW-«}{iP(y)-i?W}Mj?(y+^)-«}*{j?(y-^)-4*=o, 

where the sign of summation refers to any three arguments z, y, w, and e is any one of the 

quantities ^i, e2) ^s* 

(Cambridge Mathematical Tripos, Part I, 1896.) 



6. Shew that 



£i?i^i)= _|PJK)-#> W. 



(Cambridge Mathematical Tripos, Part I, 1894.) 
7. Prove that 

if> (2«) - p(a,,) = {j!)' (^)}-« {if> w- f> (K)P (F «- P (»»+K)}*- 

(Cambridge Mathematical Tripos, Parii 1, 1894.) 



MISC. EXS.] THE ELLIPTIC FUNCTION fp {z). 341 

8. If m be any constant, prove that 

^l r r e^{fp{z) - ft (y)} p (^) ^^^y 

" 2^J; {i?«-^i}{i?(y)-^i} ' 

where the summation refers to the values of jp («) for which jp' (^) is zero ; and the integrals 

are indefinita 

(Cambridge Mathematical Tripos, Part I, 1897.) 

9. Let 

and let (s0 {x) be the function defined by the equation 

where the lower limit of the integral is arbitrary. Shew that 

_J^Js^ ^ »^(a+y)+<^^(«) , 4>'{a-y) + 4>'{a) _ <i>'{a'¥y)-<i>'{x) 

*(A'+y)-*(«) </>(a+y)-*(«) </>(a-y)-*(«) *(«+y)-*(^) 

<»-(a-y)-<^-(:r) ^ 

(Hermite.) 

10. Shew that when the change of variables 

is applied to the equations 

rfw_. i^l — _=0, 
2iy4-l+K 

they transform into the similar equations 

2i7'+l+^f 

Shew that the result of performing this change of variables three times in succession is 
a return to the original variables |, i; ; and hence prove that if | and i; be denoted as 
functions of u by E(u) and F{u) respectively, then 

where A is one-third of a period of the fiuictions E(u) and F{u), 
Shew that • iJ^(t*)=^- j? (w ; gi> ffz\ 



12 
where ^s-^p+lgl^, 5^3= -1-^^- 216^' 



(De Brun.) 



CHAPTER XV. 
The Elliptic Functions snz, cnz, dnz, 

187. Construction of a doubly-periodic function with two simple poles 
in each period-parallelogram. 

The function ^{z\ which has been considered in the previous chapter, 
is a doubly-periodic function of z, with a single pole of the second order in 
each period-parallelogram, namely at the point congruent with the origin*. 
We shall next introduce a doubly-periodic function which differs from ^ (z) 
in having two poles, ecwh simple, in every period-parallelogram. 

Consider the series 

/(z) = S [{^ + 2mfi)i -h (2n + 1) to^}-^ - {2m^, -h (2n -h 1) ao^}-^ 

- {2r -f- (2m -h 1) ft), -h (2n -h 1) ft)^}-* -I- {(2m -h 1) ft>i + (2n -f- 1) a>»}-^, 

in which the summation extends over all positive and negative integer and 
zero values of m and n. 

When the modulus of (2ma)i -h 2nft)j) is large (and we may suppose the 
series arranged in order of ascending values of |2ma)i-h2nft),|), the terms 
of the series bear a ratio of approximate equality to those of the series 

2 [- 5 [2mwi H- (2n -h 1) ft),}-« -h z {(2m -h 1) ft), -h (2w -f- 1) ft),!"*], 



or — ^ 2 {2mft), -|- (2n -h 1) ft>a} 



i-a..--- -.,-n. 



Ol 

2ma)i +(2n-|- I)ft)al 
and these terms bear a ratio of approximate equality to those of the series 

- 22rft)i X {27nft)i -h (2n -h 1) ft)^}-*, 
which again bear a finite ratio to those of the series 

S (2mft)i -h 2nft)a)~', 
which was shewn in § 11 to be an absolutely convergent series'. 

• In the network of paraUelograma described in § 177, the poles of ^ {z) are not within the 
parallelograms, but on their bounding lines. We may however suppose the whole network 
slightly translated so as to bring the poles within the parallelograms. 



187, 188] THE ELLIPTIC FUNCTIONS sn Z, CD 2, dn z. 343 

It follows that the sertes which represents f{z) is absolutely convergent 
for all values of Zy except for the exceptional values included in the formula 

z = ma>i + (2w + 1) (Wj, (m, w, integers) 

for which the several terms of the series are infinite, and which have been 
tacitly excluded from the foregoing discussion of convergence. 

Moreover, since the terms of the comparison-series are independent of z, 
the convergence is (§ 52) not only absolute but uniform. 

By a discussion similar to that in § 176, we can shew that/(2r) is a dovhly- 
periodic function of z, whose periods are 2(Oi and 20)3 ; it is an odd function 
of Zy so that 

f{z) = -n-z); 

and its singularities are at the points 

z = m<Oi + (2n + 1) wg, 

where m and n may have any integer or zero values; these singularities 
are simple poles, with the residues + 1. There are two of these singularities 
in each period-parallelogram. 

188. Expression of tJte function f(z) by means of an integral. 

The singularities of f{z) in the fundamental period-parallelogram are, 
as we have seen, at the points z^a)^ and ^ = (Wj + oj. 

Consider now the neighbourhood of the point z = ©j. 

Writing -8^ = ft)j -I- ic, we have 

/(ft)j + a?) = — /(— 0)2 — a?), since / is an odd function, 

= — /(2ft)3--6)j — a?), since 2ft)2 is a period, 

= -/(6)2 - a?), 

from which it follows that /(oij + a?) is an odd function of x ; the expansion 
oi f{z) in ascending powers of x will therefore contain only odd powers of a?. 

Now 

f{z) » S [{a: + 2m(o^ -h (2n -h 2) to^]"^ - [2m(^ -|- (2n + 1) «,}-* 

- {a? -h (2m + 1) Oh + (2w -h 2) ©3}-^ + {(2m -h 1) ©i + (2n -h 1) <o^'% 

where the summation extends over all positive and negative integer and zero 
values of m and w. 

In this expression, replace all expressions of the form (-4 + x)"^ by 
their expansions A"^ — A~^x -h A^^a^ — . . . , x being supposed small. A term 



944 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. 

in x"^ will arise fix)m the pair of values (m = 0, n = — 1), and we thus 
have 

w 

where 5 = 2 [- [2mwi + 2wg)j}-« + {(2m -h 1) ©i + 2/wtfa}-«] . 

the summation being in this case extended over all positive and negative 
integer and zero values of m and n. excluding simultaneous zeros in the 
first term. 

If now by means of this expansion we express the quantity 

as a series of powers of x, it is found that the negative powers of x destroy 
each other ; this quantity has therefore no singularity at the point z = Wa. 

Consider next the neighbourhood of the point 2: = wi + ©j. 

Writing 2: = ©i + oja + y, we have 
/(©i -h 6), + y) = — /(— ft)i — o>j — y), since / is an odd function, 

= —/(cDi-hwa— y), since (2fth + 2ft)2) is a period. 

It follows that f((Oi + Wa + y) is an odd function of y ; its expansion in 
powers of y will therefore contain only odd powers of y. 

Now expanding/(ir) in powers of y, in the same way as /(^) was formerly 
expanded in powers of x, we find that 

/(z) l + B'y + Cy + ..., 

where 5' = S [- {(2m - 1) ©i + 2r?6)2}-» -h {2mG)i + 2nft)j}-«], 

the summation extending over all positive and negative integer and zero 
values of m and n, excluding simultaneous zeros in the second term. 

(Comparing this with the expansion of B, we have 

so /(^) = -l-5y + ay + ..., 

and, as before, the quantity 

has no singularity at the point z — fOx-\- ci>a. 

Now the points z = 6), and ^r = ©i 4- <»j are the only possible singularities 
of this quantity in the period-parallelogram ; it has therefore no singularity 
in the parallelogram, and therefore (since it is doubly-periodic) no singularities 



189] THE ELLIPTIC FUNCTIONS 811 ^, cn 2r, dn ^. 345 

in the whole -er-plane ; it is therefore by Liouville's theorem (§47) a constant 
independent o( z, say A. 

The function /(^) therefore satisfies a diflFerential equation 

Replacing B and A by new constants k and /t, we can write this in the 
form 

/'»(^) = {^,-/'(^)}{;^,-/M^)}; 

SO that, Baf(z) is zero when z is zero, 



-ris-") 



"'a.-rt dt. 



We see therefore that the odd doubly-periodic function f(z), which has 
periods 2ft)i and 2(d^ and simple poles at all points congruent with ^ = oh 
and -8^ = ft)i + 0)2, may he regarded as defined by the equation 



cnz)(]^ wji w 



where k and fi are constants depending only on cdi and Wj. 

189. The function sn z. 

The function f{z) discussed in the last two articles can be expressed in 
terms of another function, which we shall denote by sn z, in the following 
way. 

Replacing the variable t of integration by a new variable «, defined by 
the equation ks = fU, we have 

zr^fil (1 - «»)-*(! - *»«>)-* d«. 
Jo 

Now define the new function sn -e by the relation 

fifijjbz) = kanz; 



then we have 



runM 

Jo 



This last equation can be regarded as the definition of the function sn z 
m terms of its argument z and the constant-parameter k, which is called 
the modulus ; it is analogous to the definition of the function sin z by the 
relation 



rwnz 

= / (1 -«»)-* ds. 



346 TRANSCENDENTAL FUNCTIONS, [CHAP. XV. 

From the equation 

fif(jiz) = & sn ^, 

it is clear that the function sn z has the same general properties as f{z)y 
namely, it is an odd one-valued doubly-periodic function of z, with two poles 
in each period-parallelogram, the distance between the poles being half of 
one of the periods. The two periods will be connected by a relation, as they 
depend only on the single constant h 

190. The functions en z and dn z. 

We now proceed to introduce two other functions, either of which may 
be regarded as bearing to the function sn^ a relation similar to that which 
the function cos z bears to sin z, 

Csaz 

Since z^\ (1 - «»)-* (1 - k"^)-^ ds, 

Jo 

dz 

we have -r? c = (1 — sn* z)~^ (1 —k^ sn' z)~^, 

a{snz) ^ ^ ^ 

or -T- (sn -^) = (1 - sn' z)^ (1 - A:" sn» z)K 

Now sn ir is a one-valued function of z, so its derivate must be also a one- 
valued fimction. It follows that 

(l-sn'^)*(l-A'sn«2r)* 

can have no branch-points (§ 46), considered as a function of z ; and therefore 
either 

(a) Elach of the quantities (1 — sn'^r)* and (1 — Ar'sn'^r)* is a function of z 
which has no branch-points, or 

(/8) The functions (1— sn'z)* and (1— A'sn'^)* have branch-points, .but 
are such that their product has no branch-points. 

Now the alternative (fi) could be true only if the functions (1 — sn' z)^ and 
(1 — Ar'sn'^)* had their branch-points at the same places ; but this is not the 
case, since (1— sn'^)* has branch -points at the places when sn'2: = l, and 
(1 — A'sn'^)* has not. The alternative ()8) being thus ruled out, we see that 
the alternative (a) must hold. 

If now we write 

en 2: = (1 — sn'^:)*, 

dn;r = (l — i'sn'-e)*, 

where it is supposed that each of these functions has the value unity when 
sn z is zero, then since en z and dn z have no branch-points, and have definite 
values at the point 2: = 0, it follows that the functions en z and dn z are one- 
valued functions of z. 



190, 191] THE ELLIPTIC FUNCTIONS 811 ^, cn z, dn z. 347 

They obviously satisfy the relations 

an^z + cn^z= 1, 
l^sn^z + dn'2:= 1. 

The functions sn z, en z, dn z are often called the Jacobian elliptic 
Ainctions, 

The function cos z is in the same way a one- valued function, although the occurrence of 

the radical in (1 -sin'^)' might lead us at first sight to suppose that it possessed branch- 
points. 

191. Expression of en z and dn z by means of integrals. 

We shall next find, for the functions en z and dn z, integral-expressions 
similar to that found in § 189 for sn^^. 

Diflferentiating the equation 

cn*z = 1 — 8n*z, 

we have en -^ -y- en -? = — sn .8^ en z dn -s^, 

dz 

so -7- en -? = — sn -g' dn z 

dz 

= - 1(1 - en- z) (*'« -h k" cn« z)]^, 
where &'* = 1 — A;*. 

Thus ifcn^ = ^ we have 

d^ = - (1 - <»)-* (*'« + h^t")-^ dt, 
and therefore (since en ^ = 1 when z = 0) 

z^r (l-.t«)-*(A'»-hifc»t')-^(ft. 

In the same way we can shew that 

J- dn -? = — A^ sn z en z, 
dz 

and z=r (l-^)-*(t«-A/»)-*d«. 

J dm 
Example 1. If cs 2»cn ^/sn z^ shew that 

J C8« 

Example 2. If sd 2 -> sn zjdn ^ shew that 



- {"^ \\ - h'H^yk (1 +i:»^)-i efe 



i 



348 TRANSCENDENTAL FUNCTIONa [CHAP. XV. 

192. The addition-theorem for the function dn z. 
We shall next shew how to find dna:, where 

in terms of the sn, en, and dn, of y and z : the result will be the addition^ 
theorem for the function dn. 

Suppose that y and z vary, x remaining constant, so that 

^^ — — 1 

* Introducing new variables u and v, defined by the equations 

w = en ^ en y, 
t; = sn ^ sn y, 



we have 



or 



dv 1 . ^ d^ 

, -T- sn 2: en V dn V + sn V en 2r dn -er -y- 

dv _d£^ ^ ^ ^ ^ dy 

du du \ \ az 

T- — en z sn y dn y — en y sn -8^ dn ^ -1- 

dv __ sn 2r en y dn y — sn y en ^ dn ir 
du Gnysazdnz—cnzsnydny' 



From this we obtain the equations 

(-T-) - 1 =A:*(sn'y — 8n*2r)'(cny sn ^dn 2r — cn^rsny dny)"*, 

dv 
v^u-T- = (sn y en y dn ^ — sn 2: en ^ dn y) (en y sn j dn 2: - en 2: sn y dn y)""^ 

(-r-j ■~(^"'^;7~) =(sn'y--sn'^)'(cny sn^dnz — cn2:8n ydny)"*, 
and consequently 

L /"^V -. 1 - /"^ V ( - ^^V 

i^Kdu) k'^Kdu) V ^'dul' 



or 



*■(«-"*■)'—*•■©■■ 



This equation is the equivalent, in the new variables, of the equation 

dz - 

dy" 

It is a differential equation of Clairaut's type, and its integral is therefore 

ifc» (t; - -Mc)* = 1 - A: V, 
where c is an arbitrary constant. 



192] THE ELLIPTIC FUNCTIONS sn ^, CH Zy dn z, 349 

Thus the equation 

A» (sn 2r sn y — c en -^ en y)* = 1 — AV 

must be equivalent to the equation 

where c is some funetion of x. 

To determine c in terms of x, put y = ; then we have 

ifc«c* en* a; = 1 - Jk V, 
which gives c* = dn"'^; = dn~* (z + y). 

Now the integral equation can be written in the form 

c» (1 — Jt» -f jfcj en' y en' ^) — ick^ sn y sn 2: en y en ^r + (A:* sn' y sn' 2r — 1 ) = 0. 

Solving this equation in c, we have 

_ A^'sny sn 2: en y en ^ t {Ar* sn'y sn'^r en'y en* ^—(1— Ar'-I-A:' cn'yen*2:)(A:'sn'y sn'^—1 )}* 
" 1 — A* -h A* en* y en* ^ 

_A:*snysn2renycn^±dnydn^ 
^^ ^" 1-A* + A;*en*yen*^ ' 

Sinee 
A:* sn* y sn* 2: en* y en' ^ — dn*y dn* = (1 — A;* + A* en* y en* 2:) (A:* sn* y sn* ^ - 1), 
this equation ean be written 

A:* sn* V sn* ^ — 1 



or 



4* sn y sn 2r en y en 2r T dn y dn ^ * 

, . . ±dnydn^iA:*snysn2:enyen^ 

an \Z + y) = = — ^i — i z . 

^ ^' 1 — A:»sn*ysn*z 



The two ambiguities of sign in this equation remain to be decided. 
Taking ^r = 0, it is seen that the first ambiguous sign must be + ; so 

J . . dn ?/ dn ^ + A^ sn V sn 2: en V en ^ 
dn(2r + y)i= — ^ -:=— — f — ^ . 

^ ^ 1 — &*sn*ysn*-2r 

Now suppose that y is a small quantity ; expanding both sides in ascend- 
ing powers of y, and retaining only the terms involving the first power of y, 
we have 

dnz-^-y-^dnz^dnz ±I^8nzcnz, 

Sinee -j- dn ^ = — i* sn 2? en jg, 

dz 



360 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. 

it is clear that the ambiguous sign must be — . We thus finally obtain the 
addition'theorem for the function dn, namely 

, , . dn-edny— Ar^sn^sny cn^cny 

Example 1. Shew that 

^' ^ ^' 1 - ifc* an* y sn* « 

Example 2. Prove that 

, J „ 2dn2^ 

l+dn2z 



193. The addition-ikeorems for the functions sn z and en z. 
To obtain the addition-theorem for the function sn ^, we have 

sn(ir + y)=±j^{l-dn«(«+y))» 

Substituting for dn {z + y) from the result of the last article, this equation 
after some algebraical reduction gives 



sn 



. . _ sn ^ en y dn y + sn y en 2: dn -gf 
(^ + y) - ± - l^^Jc^^^^Jsn^ • 



On putting y = in this formula, it is seen that the ambiguous sign is -h ; 
we thus obtain the additton-theorem for the function sn, namely 

I V _ sn -8^ en y dn y + sn y en 2r dn -? 
sn(^ + y)- —-,^^——— . 

Similarly for the function en z we obtain the addition-theorem 

. . _ en ^ en y — sn 2r dn ir sn y dn y 
^""^^■^^^ 1-A;«sn»^sn^y * 

These results may be regarded as analogous to the addition-theorems for 
the circular-functions, namely 

sin (-3^ -h y) = sin z cos y + cos z sin y, 

cos (-8^ + y) = cos z cos y — sin ^ sin y, 

to which, indeed, they reduce when k is put equal to zero. 

Example 1. Prove that 

sn(z4-y)sn(«-y)=:; — rs — 5 — o ^ > 

Example 2. Shew that 

, 1 - en 2jj 

1 H-dn 2« 



193-195] THE ELLIPTIC FUNCTIONS SD Z, CU Z, du Z, 351 

194. The constant K, 

We shall denote the integral 

r(i-eo-*(i-jfc»e«)-*ctt 

by if; it is clearly a constant depending only on the modulus k. The 
ambiguity of sign in the radical will be removed by the supposition that at 
the lower limit of integration the integrand has the value 1. 

From the equation 






z 

'o 



we see that sn ir= 1, 

and hence en if = (1 — sn^ K)^ = 0, 



Example, Prove that 



dnir=(l-ifc»8n«£')* = A?'. 






196. The periodicity of the elliptic functions with respect to K, 

It will now appear that the constant K is intimately connected with the 
periodicity of the elliptic functions sn z, en z, dn z. 

For by the addition-theorem, we have 

sn -g: en X* dn X* 4- sn ^ en z dn 2r en z 



sn{z + K):= 



Similarly cn(z + K)=='-k' ~ 



l-k'sn'^zsn^K dnz' 

, sn^: 



and cln(^-|-ir) = ^ 



z 



Hence sn {z + "IK) = j^ / ^ jl = - sn £, 

and similarly en {z + IK) = — en -3^, 

dn(2: + 2iO = dn2r; 
and finally sn (z + 4ir) = — sn (^ + 2ir) = sn z, 

en {z + ^K) = en z, 

dn {z + 4i0 = dn z. 



352 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. 

This 4iK is a period for the functions sn z and en z, arid 2K is a period for 
the function dnz. 

Example. If cs 2 = en z/an z, shew that 

cszca(K-z)^t, 

196. The constant K', 

We shall denote the integral 

The ambiguity of sign in the radical will be removed by supposing that 
at the lower limit of the integration the integrand has the value 1. 

Write « = (1 - ib"^)-*. 

Then (i-^)-*=^(l-A:V)i, and (1-^^)"* = ^^-^^*^ 

and d« = (1 - k'H^)'^MHdt. 

1 

Therefore £" = - i\ (1 - «»)-* (1 - h^^Y^ ds, 

1 

and so ' K-\- iK' = [*(! ~ «>)-* (1 - A^^)-* ds, 

Jo 

or m(K + iIC)^Ty 

whence dn(ir + iZ')=0 and en (ir + tir)= ± ^. 

To determine the ambiguous sign in the last equation, we observe that 
the sign of i must be understood in the light of the relation 

(l-s')-*=^(l-A;'»<')». 
which was used in the transformation ; putting 

« = sn(Z + iir') = p «=1. 

ihf 
And so en {K + iK) = - -^ . 



Example. Shew that cni(ir+iJr')=(l-t)(^) . 



196-198] THE ELLIPTIC FUNCTIONS sn 2r, en z, dm, 363 

197. The periodicity of the elliptic functions vrith respect to K + iK', 

The quantity K introduced in the last article is of importance in 
connexion with the second period of the functions snz, cn-gf, dnir. 

For by the addition-theorem, we have 
/ . IT . -ET/x sn<^cn (^+tX')dn(^ + tir') + sn(-K' + tZOcn<^dn^ 

Sn (j^ + iL + iJtL ) = ?— ;; j-i TTtt ^TrTl 

_ dn-er 
A; en 2:' 

ik' 1 
Similarly en (^ + ^ + iK') = - -r- , 

xkj sn £ 

and , dn (^ + iT + iK') = . 

^ GHZ 

By repeated application of these formulae we have 

sn (z + 2^^ + 2tZ') = - sn-g:, 
en (^ + 2ir + 2iK) = en z, 
dn(-^ + 2ir + 2iir) = -dnr, 

aod fsn(^ + 4ir+4tZ0=sn^, 

cn(^-|- 4^+ UK') = Gazy 
dn(^-|-4J5r+4tZ') = dn2r. 

Hence it appears that the function en z admits the period 2K + 2iK'y and 
the functions sn z and dn z admit the period 4}K + 4iir'. 

198. The periodicity of the elliptic functions with respect to iK. 
By the addition-theorem, we have 

sn (^-1- iZ') = 8n (z+ K+ iK'-K) 

^ sn (z ■¥ K + iK')cn K dnK - mK en {z '\- K + iK)dn(z + K '\'iK') 

l-k'sn^K sn^ (z + K-^-iK) 
1 
^ ksnz* 

Similarly we find the equations 

^ k snz 

dn (z +iK) = — i . 

snz 

w. A. 23 



354 TRANSCENDENTAL FUNCTIONa [CHAP. XV. 

By repeated application of these formulae we obtain 

sn (z + 2iK') = an z, 
en {z + 2iK') = — en -gf, 
dn{z + 2iK')^--dnz, 

and (Bn(z + ^K')^snz, 

en (z + 4dK') = en z, 
dn (z + 4dK') =^ dn z, 

80 that the function sn z admits 2iK' as a period, and the functions en z and 
dn z admit 4dK' as a period. 

199. The behaviour of the functions sn z, cnz,dnz, at the point z = iK\ 

For points in the neighbourhood of the point 2f = 0, the function saz can 
be expanded by Taylor's theorem in the form 

snj? = 8n0 + 58n'0 + 2^sn"0 + ..., 

where accents denote derivativea 
Since sn = 0, 

sn'0 = cnOdnO = l, 
8n''0 = 0, 



the expansion 


becomes 


sn'"0=.-(l + A;»), etc. 
8nz ^ z "1(1 -{- 1(^) z^ + ... . 


Hence 




cnj? = (l — sn'z)* 

= 1—2^ +..., 


and 




dn^ = (l-*»sn«^)* 


and therefore 








sn(^ + tX) = -^i-- 



=5^{l-l(l+A.)^+...f-^ 



1 . i + h' 



199-201] THE ELLIPTIC FUNCTIONS 811^, cn z, dn z. 365 

— i 2k^ — 1 
and similarly en (z + iK") = nr- H gjr- i? + ... 

i 2 — Jfc* 
and dn(-2r + i-K'') = --+ — f— i« + .... 

It follows that at the point z = iK\ the functions sn z, cnZfdnz have simple 

poles, with the residues 

1 _i 

k' k' *' 

respectively. 

200. General description of the functions sn ^, en z, dn ^. 

Summarizing the foregoing investigations, we can describe the functions 
sn z, en Zy and dn z, in the following terms. 

(1) snz is a one- valued doubly-periodic function of z, its periods being 
4iK and 2iK\ Its singularities are ^t all points congruent with z^iK' 
and z = 2-ff' -f iK* ; they are simple poles, with the residues Ar* and — k"^ 
respectively ; and the function is zero at all points congruent with ^ = and 
z = 2K, 

It may be obeerved that no other function than sn z exists which fulfils this description. 

For if </>(;;) be such a function, then 

^ {£) - sn « 

has no singularities, and so by Liouville's theorem is a constant independent of z ; but it is 
zero when 2=0, and therefore the constant is zero ; that is. 

When ](^ is real and positive and less than unity, it is e€wily seen that K 
and K' are real, and sn z is real for real values of z and purely imaginary for 
purely imaginary values of z, 

(2) en ^ is a one- valued doubly-periodic function of z^ its periods being 
4iK and 2K-\- 2iK\ Its singularities are at all points congruent with z^iK' 
and z=i2K -{-iK* \ they are simple poles, with the residues ik^^ and —ik~^ 
respectively ; and the function is zero at all points congruent with z==K and 
z^ZK. 

(3) dn ^ is a one-valued doubly-periodic function of z, its periods being 
2K and 4dK\ Its singularities are at all points congruent with z = iK' and 
z = ZiK' ; they are simple poles, with the residues — i and + 1 respectively ; 
and the function is zero at all points congruent with z = K + iK' and 
z^K^2iK\ 

201. A geometrical illustration of the functions snz, cnz, dnz. 

The Jacobian elliptic functions may be geometrically represented in the 
following way. 

Let the position of a point, on the surface of a sphere of radius unity, be 
defined by (1) its perpendicular distance p from a fixed diameter of the 

23—2 



356 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. 

sphere, which we shall call the polar axiSy and (2) the angle y^ which the 
plane through the point and the polar axis (the meridian plane) makes with 
a fixed plane through the polar axis. 

Then if ds denotes the arc of any curve traced on the sphere, we clearly 

have the relation 

(day = p« {dylry + (1 - p»)-' (dp^. 

Let a curve (SeiflFert's spherical spiral) be drawn on the sphere, its 

defining-equation being 

dy^ = kds, 

where A? is a constant. We have therefore for this curve 

and so if « be measured from the pole, or point where the polar axis meets 
the sphere, we have 

« = f ' (1 - /)•)-* (1 - A;»p»)-* dp, 
Jo 

or p = sn «, 

the function sn being formed with the modulus k. 

The rectangular coordinates of the point s of the curve, referred to the 
polar axis and an axis perpendicular to it in the meridian-plane, are p and 
(1 — p')*, and can therefore be written sns and cns\ while dn* is easily seen 
to be the cosine of the angle at which the curve cuts the meridian. Hence 
if K be the length of the curve from the pole to the equator, it is obvious 
that sn s and en s have the period 4jK', and dn s has the period 2K. 

202. Connexion of the function sn z with the fwnction p (z). 

We shall now shew how the functions considered in this chapter are 
related to the elliptic function of Weierstrass. 

Let ei, ej, e^ denote the quantities ei, ^, ^, taken in any order. 

Li the integral 

z^\ I (a? - e^~^{x - ^)-*(a? - e^-^dx, 

let the variable of integi'ation be changed by the substitution 

ei-ej 



a: = e^ + 



Thus 



z= {l-t')'^{(ei-ej)-b(ej--et)t'}'^dt, 

Jo 

or {ei - 6^)* -8^ = (1 - «•)H^ (1 - l(^t^)-^ dt, 

Jo 



202, 203] THE ELLIPTIC FUNCTIONS sn z, cn z, dn z, 357 



where A* = 



e*-e,- 



This is clearly equivalent to the equation 

We thus obtain the result that the function ^{z)yf(yrmed with cmy periods y 
can he expressed in terms of the function snz by the equation 

the function sn being formed with the modulus 

\ei - CjJ 
Example, Shew that this relation can be written in either of the forms 

and p(,)=^*-^^5!i(fiZ^}. 

l-dnM(^-«,)*4 

203. Expansion of snz as a trigonometric series. 

Since sn^r is an odd function of ^, admitting the period 4ir (which we 
shall for our present purpose suppose to be real), it can by Fourier's theorem 
be expanded in a series of the foim 

, . irz , 27r£^ 7 . oirz 
sn^r = 6, sm 2^ + 6, sm 2^ + 6, sin 2^ + ... , 



1 f*^ 
where (§82) ^'' " ^ I s^^ ^ sin 



2K 

This expansion will (§ 78) be valid for all points in the ^-plane contained 
in a belt parallel to the real axis and bounded by the lines whose equation is 

Imaginary part of 2r = ± iK\ 

since within this belt the function sn z has no singularities. 

We have now to evaluate the integrals 6^. We shall follow a proof due 
substantially to Schlomilch. 

Let OARSCBQPO be a figure in the plane of a variable t, consisting of 
the rectangle whose vertices are the points 

0(t = 0), A(t^ 2K)y C(t^2K+ 2iK')y B{t = 2iK% 

with a very small semi-circular indentation PQ around the point t = iK\ and 
another small semi-circular indentation R8 round the point t = 2K -f iK\ 



358 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. 

Consider the integral 

I sn < « *^ (ft, 
taken round this contour. 

Since the integrand is regular everywhere in the interior of the contour, 
we have (§ 36) 

\ +1 +1 +( +( +f +f +f =0. 

J OA Jab J es j so J cb J bq J qp J po 

Consider first the integral along the semi-circular indentation QP, 
Writing t = i-fiT' + Re^, we have 



Jqp Jw 



irwt /.-« nrJT inr 

qp 

2 

w irw 






snz 



2 

IT 



i ^T^ r"2 
= T « *^ / (1 + positive powers of R) dO 



ir% - 



a 

rvK' 



= — r ^ ^^ J when R tends to zero. 
k 

Similarly we have 

imt • rwK' 



snte^dt^i-iy^e' ^ . 
Jbs k 



Since sn (z + 2%K') = sn z, we have 

J CB J OA* 

and since sn (z + 2iQ = — sn ir, we have 

f =(-iyf , and f =(-l)'f . 

J^tB J PO J 80 J BQ 

We thus have 

Now equate to zero the imaginary parts of this equation. Since 

iiyt 

sate^^ dt 



203] 



THE ELLIPTIC FUNCTIONS anz, CD 2, dn^. 



359 



is real when t is purely imaginary, there is no imaginary part arising from 



JBQ JPO 



Therefore 



nrJST' 



nrJT 



(l^e^)j^mtBm^dt=le'^{l^(^iy}. 



2K k 



Writing 
this equation gives 






IT 



{\-<f)KK='j^<f{\-{-\r] 



or 
and 



6y = if r is even. 



Thus finally we have the expansion of sn ^ as a trigonometric series, 



sn^ 



_ 27r / g* irz g* 



Example, Prove that 



. ^irz (fi . ^irz 



A 



• • • I • 



Stt ( q^ nz , q^ Zirz . o* bnz ] 

^^=2?tri^"^2Z + ii73«>«2^ + l-^co8^ + ..,|. 



Miscellaneous Examples. 



1. Shew that 



2. Shew that 



3. Prove that 



4. Prove that 



6. Prove that 



6. Prove that 



ens 



rdn* 



dt 



z 



/"cn z 



{l±cn(,+y)}{l±cn(.-y)}=J^^|Bg_ 



dn*«= 



ir^-fdn 2g -hitr»cn2g 
Hrdn2« 






cn? 



360 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. 

7. Shew that 



^ , dnz-cnz 



t^+dnz-k^cm" 



8. Shew that 



8n(*+iJr)=(l+i') S-(1-^)SD«, ' 



/ 



sn 

9. Prove that 

8111 [srn-i {sn («+y)}+8m-i {an («-y)}]=^j__-j_-^. 

10. Shew that 

r • 1 f / . M w / xn cn'y— sn*ydn*« 

co8[sm-i {aD(r+y)}~8m-M8n(^~y)}]= i_^aD»IaD«y * 

11. Shew that the quarter-periods K and iK' are solutions of the equation 

where «=i!*. 

12. Shew that the quarter-periods K and tX" are Legendre functions of the argument 
(l-2ife»), of order -J. 

13. Shew that 

en jScny sn O— y)dn a+cn y en a sn (y- a) dn /9+cn a en j8 sn (a- /3) dn y 

-l-sn — y) sn (y - o) sn (a— /3) dn o dujS dn y =0. 
(Cambridge Mathematical Tripos, Part I, 1894.) 

14. l{u+v+r+s=Oy shew that 

i;^ sn t« sn V en r en « - /r> on t£ en t7 sn r sn « - dn t« dn v+dn r dn «=0, 

iP^ sn t« sn i;- iP^ sn r sn «-|-dn i« dn V en r en « - en tt en vdn r dn tfasOy 

sn t^sn V dnrdn «~dn ii dn vsnrsn «+cn r en «-ont«cni7sO. 

(H. J. S. Smith.) 
16. Shew that, if a>x>fi>yy the substitutions 

x~y=(a— y)dn^t« and ar-y=(3-y)dn"*», 

where it* ■« (a — jS) (o - y) ~ *, reduce the integrals 

j {{a-'X){x- p){X'' y)}-^ dx and | {(a-x)(a?-/9)(4?-y)}-*d:a? 

respectively to the forms 2tt (o - y) " ^ and 2v (a - y) " * ; and deduce that, if w + v = JT, 

1 - sn^ u - sn* v+ifc* sn* u sn* v=0. 

From the substitution y= (a- .ir) (a?-/3) («-y)~^ applied to the integral above with the 
limits p and a, obtain the result 

W IT 

1%! cos* ^ + 6i sin* d) - * c?d = /V* COS* ^ + ft* sin* d) - * <W, 

where o^, 6] are the arithmetic and geometric means between a and b. 

(Cambridge Mathematical Tripos, Part I, 1895.) 



w 






MISC. EXS.] THE ELLIPTIC FUNCTIONS Sn2r, CU Z, du Z. 361 

16. Shew how to exprees 



I 



as an elliptic integral of the first kind, in the case when both quadratic expressions have 
imaginary linear factors. 



If z^r{(x+l)(x^+x+l)]-idx, 



express x in terms of z by means of Jacobi's elliptic functions. 

(Cambridge Mathematical Tripos, Part 1, 1899.) 

17. The difierent values of z satisfying the equation cn3z=a are z^, ^, ... z^. 
Shew that 

9 

3iH n en Zr+f* U cn«,.=0. 

(Cambridge Mathematical Tripos, Part I, 1899.) 

18. Shew that 

en « 2ir ( q^ 1FZ q^ ZirZ . q^ birz 1 

J = r^ {r" — C0S2rp.-T-^---iC0S^r^+^^— iCOB-r-p— ...}-. 

auz JcK [l-q 2K 1-^ 2K 1-^ 2K j 

19. Prove that 

k'anz 2ir 



dnz 
20. Shew that 



27r f V • «■« Q^ ' 37r2 . q^ . 6rrz ] 

= 15- (i +y «« 2Z - d^ """ 2^ + 1 +? ""• iz - •••; • 

J ir 2n ( q itz ^ (fi 2ttz . \ 



21. Prove that 



sn' 



/l+ifea/TrX 1« /»r\8\ V • ^« 

fl-Hr>/tr\ 3^ /7r\»l V . 
"*"t 2;fc3 \^2iS:; 2/fc3 \^2iry / 1-^ ^ 



2K 

Zirz 

sm — => 

2K 



biFZ 

sm 



"^t 2it3 V2i:; 2)E3 V2ir; j i -^^" 2ir 

(Cambridge Mathematical Tripos, Part II, 1896.) 

22. Shew that 

*« sn2 j= p (a - xK') + Constant, 

where the Weierstrassian elliptic function is formed with the periods 2K and 2xK'. 

23. Shew that the differential equation 

g={Ji*8n«*-i(l +*»)}« 
admits the general integral 

i* = {8ni(C-«)cni(C-a)dni(C-«)}-*{i4 + i9sn«i(C-«)}, 
where A Mid B are arbitrary constants, and C^2K+%K\ 



CHAPTER XVI. 
Elliptic Functions; General Theorems. 

204. Relation between the residues of an elliptic function. 

In this chapter we shall be chiefly concerned with properties of more 
general elliptic functions than the special functions p (z), sn z, en z, and 
dn^, which have been discussed in the two preceding chapters. 

We shall first shew that the sum of the residues of any elliptic function, 
with respect to those of its poles which are situated in any period-parallelogram, 
is zero. 

For let f(z) be an elliptic function, and let icji and 2(0^ be its periods. 
The sum of the residues is, by § 56, equal to the integral 

taken round the perimeter of the parallelogram. 

Now in this integral, any two elements f(z)dz corresponding to congruent 
line-elements dz on opposite sides of the parallelogram, are equal in magnitude 
but opposite in sign, and therefore destroy each other. Hence the integral is 
zero, which establishes the theorem. 

The number of poles or zeros of an elliptic function contained within 
a single period-parallelogram is often referred to as the number of irreducible 
poles or zeros. 

205. The order of an elliptic function. 

We shall next shew that if c is any constant and f(z) is an elliptic 
function, the number of roots of the equation 

contained within a period-parallelogrami depends only on f(z), and is inde- 
pendent of c, and is therefore equal to the number of irreducible zeros, amd also 
to the number of irreducible poles. 



204-206] ELLIPTIC FUNCTIONS i GENEBAL THEOREMS. 363 

For the difference between the number of zeros of the function 

m - 

and the number of its poles, contained within the parallelogram, is (§ 60) 
equal to the value of the integral 



!«•]■ 



/'W 



dz 



2inj f{z)''C 

taken round the perimeter of the parallelogram. But if P and Q are two 
points congruent with each other, situated on opposite sides of the parallelo- 
gram, then the elements /' {z) [f{z) — c]"^ dz arising from P and Q are equal 
in magnitude but opposite in sign, and so destroy each other. The integral 
is therefore zero; that is, the number of zeros of the function f(z)'-o 
contained within the parallelogram is equal to the number of its poles, i.e. to 
the number of the poles of 'f{z) ; but this latter number is independent of c, 
which establishes the theorem. 

The number of irreducible poles or zeros of an elliptic function is called 
the order of the function. It must be noted that a zero or pole, which is 
multiple of order n in the sense of " order " defined in §§ 39, 44, must be 
counted as n zeros or poles for the purposes of this definition of " order." 

The order is never less them two; for if an elliptic function had only 
a single irreducible simple pole, the sum of its residues within any period- 
parallelogram would not be zero, contrary to the theorem of the last article. 
This explains why the functions discussed in the two preceding chapters, 
which are of order two, are the simplest elliptic functions. 

206. Expression of cmy elliptic function in terms of ^ {z) and jp' {z). 

We shall now shew how any elliptic function can be expressed in terms 
of the Weierstrassian elliptic function which has the same periods. 

Let f{z) be any elliptic function, and let ^{z) be the Weierstrassian 
elliptic function with the same periods 26t>i and 2q)s ; and let f^ {z) be the 
derivate of ^{z). 

First, we can write 

f{z)-\ [fiz) + /(- z)] + Y^l^I^ ^' (^). 

Now the functions f{z)+f{—z) and {/(^) — /(— -sr)} p'~*(^) are even 
elliptic functions of z\ let {z) denote either of them : we shall now express 
^ {z) in terms of (p {z). 

Since <^(^) is an even function, it follows that if a be one of its zeros 
in the fundamental period-parallelogram, then another of its zeros in the 
parallelogram will be congruent to — a : its iiTeducible zeros may therefore 
be arranged in two sets, say (h» (h>'"CLny and zeros congruent to 



364 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI. 

Similarly its poles can be arranged in two sets, say 61, 62, ... 6n» and poles 
congruent to — 61, — 6s, ... — 6n« 

Now form the quantity 

~4> (^) { J> (^) - P (6i)} {f> (^) - jf> (6,)} . . . {JP W - i? (6n)} ' 

where if one of the quantities Or or 6^ is zero, the corresponding factor 
{pC-^) — j?(M} or { JP (^) -" j? (M) is to be omitted. 

This quantity is a doubly-periodic function of ^r ; it clearly has no zeros 
or singularities in the interior of the parallelogram, except possibly at ^^ = 0, 
and therefore either it or its reciprocal has no singularities in the interior 
of the parallelogram, and so has no singularities in the entire plane. It must 
therefore by Liouville's theorem be a constant independent of z. 

Thus 

A (z^ - Constant x {<^^^>^ P fa)l W (^) - <f> («.)! » » » {P (^) - P K)) 
<^(.)- Constant x {p(,)_p(i^)j lj>(.)-|>(6,)} ... {f> (.) - jXtn)}" 

The quantities {/(^) +/(--?)} and {/(^) -/(--?)} {f>'(^)l-* can thus be 
expressed as rational functions of ^{z)\ and thus we obtain the theorem 
that any elliptic function can he expressed in terms of the Weierstrassian 
function formed with the same periods, the expression being linear in ff {z) 
and rational in jf>(^). 

Example. Shew that 

snzcTkZ^uz^ ^k"^ ^{z- iK'\ 

where the function p' {z - iK') is formed with the periods 2K and 2iK'<, 

207. Relation between any two elliptic functions which admit the same 
periods. 

We shall now shew that an algebraic relation exists between any two 
elliptic functions whose periods are the same. 

For let f(z) and <l>(z) be the functions. Then by the last article, f{z) 
and (f> (z) can be expressed rationally in terms of jp (z) and jp' (z). Eliminating 
jp(z) and p'(z) from the three equations constituted by 

p''(z) =- i^ip^iz) ^ g,p(z) - g, 

and these two relations, we have an algebraic relation between /(^) and <(> (z) ; 
which establishes the theorem. 

It is easy to find the degree of this equation in / and <}>. For if / be an 
elliptic function of order m, and if ^ be of order n, then each value of/ 
determines m irreducible values of z, and each of these determines one value 



207, 208] ELLIPTIC FUNCTIONS ; GENERAL THEOREMS. 365 

of <^ : so to each value of / correspond m values of <^, and similarly to each 
value of <t> correspond n values of /. The equation is therefore of degree m 
in ^ and n in/ 

Thus IP {z) is of order 2, and jp' (z) of order 3. The relation between them, namely 

rW=4pW-(73if>(^)-5r3, 
should therefore be of degree 2 in fp'iz) and 3 in (p {z) — as in fact it is. 

An obvious consequence of this proposition is that every elliptic function 
is connected ,tuith its derivate by an algebraic equation. 

Example. If ty u, v are three elliptic functions of the second order, with the same 
periods and argument, shew that there exist in general between them two distinct relations 
which are linear with respect to each of them, namely 

Atuv-\-Buv+Cvt+Dtu+Et+Fu-^Ov+H=0y 

A'tuv+B'uv+C'vt+iytu-\-E't+F'u+O'v+E'==0y 

where A, B, ,,, , H* are constants. 

208. Relation between the zeros and poles of an elliptic function. 

We shall now shew that the sum of the affixes of the irreducible zeros of am, 
elliptic function is equal to the sum of the affiles of its irreducible poleSy or 
differs from this sum only by a period. 

For it f(z) be the function, and 2a)i and 2a)8 its periods, the diflFerence in 
question is (§ 59) equal to the integral 

1 [ zf{z)dz 
27rij f{z) 

taken round the perimeter of the fundamental period-parallelogram. This 
can be written 



or 



or 



JL r f- {zfS') _ (20,. + ^)/' (26), + ^) ] , 
27riLJo t/{^)" /(2a„ + ^) J*" 

+ p L '/'(^ + (2a>, H- y (2., -M) ^1 



or 



^{-^^'m^^'^'M 



or 2«^~^**^'®S^"'"^*'''"Sl}. 



366 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI. 

and as log 1 is zero or some multiple of 27rt, the last expression must be 
either zero or some quantity of the form 

A multiple of 2a>i + A multiple of 2a)a, 

ie. a period This establishes the theorem. 

Example, If F{£) is an elliptic function, for which ^, ^, ... are the irreducible poles, 
and Ai, A^, ... the corresponding residues, and it f(z) is a one-valued function without 
singularities in the parallelogram, shew that the integral 

taken round the period-parallelogram, is equal to 2A^f(z^). 

(Cambridge Mathematical Tripos, Part II, 1899.) 

209. Thefimction f (z). 

We shall next introduce a function f (5), defined by the equation 

with the condition that ^{z) — z^^isto be zero when 2r = 0. 

Since the infinite series which represents |> (z) is uniformly convergent, it 
can be integrated term by term ; we thus have 

f (^) = - /" [ir> + S {(-r - 2mfih - 2no>,)-* - {2nm, + Sno^)-*}] dz 

= -2^* + S {(-r - 2ma>i - 2na>^)-^ + (2ma}i -{■ 2na),)-^ -f z (2mcih + 2nw,)"^}, 

since the condition, which ^(z) has to satisfy at2r=:0, is satisfied by this 
choice of the constant of integration. The summation is, as usual, extended 
over all positive and negative integer and zero values of m and w, except 
simultaneous zero values. 

When I 2ma>i 4- 27iq>, | is large (and we can suppose the series arranged in 
ascending order of magnitude of | 2mo)i + 2na)s i)> the quantity 

(z — 2mcDi — 2na)8)'"^ -f (2ma>i -f inaj^"^ -f z (2m<0i -f 2na)3)~* 

bears a ratio of approximate equality to the quantity 

- z^ (2?nft)i -i- 2yw»a)-». 

The series which represents ^(z) can therefore be compared with the 
series S (2nui>i -f 2n<»,)~*, and hence we see that it is absolutely convergent, 
except at the singularities z = 2ma)i 4- 2na>2, ^^^ that the convergence is 
uniform. 

It is evident from the series that at its singularities z = 2mfi)i -f 2wg)2, the 
function ^(z) has simple poles with residues unity; and that ^(z) is an odd 
function of z. 



209-211] ELLIPTIC functions; general theorems. 367 

The function C (') niay be compared with the function cot ;;, whose expansion is 

cot«=«->+ 2 {(«-wwr)"* + (i?Mr)->}. 

The equation t- cot z = - cosec* z 

corresponds to the equation 

210. The quasi-periodicUy of the function f(«). 
Since !>(•* + 2o),) = |> {z), 

we have _ f (^ + 2a),) = ^ ? (4 

or f(^ + 2w,) = ?(«) + 2i7„ 

and similarly K{^-^ ^Wa) = ?(-?) + 2i7„ 

where i/i and 172 are two constants introduced by integration. 

Writing z^ — toi and 2r= — ©j in these relations respectively, we have 

f (oh) = f (- «.) + 217, = - f (a>0 + 2i7„ 

ir («.) = f (- ««) + 2172 = - ?(a>2) + 2i;„ 
whence Vi^K (®i)> 

which determines the constants rji and 17,. 
If a:+y+«=0, shew that 

{fW+f(y)+fW}*+r(^)+rcy)+rw=o. 

(Schottky.) 
This result may be regarded as the addition-theorem for the function ( {z), 

211. Expression of an elliptic function, when the principal part of its 
expansion at each of its singularities is given. 

Lei f(z) be any elliptic function, with periods 26)i and 2a>2. Let its 
irreducible singularities be at the points z=^a^, ctj, ... On; and let the 
principal part of its expansion near the point ak be 



z-au {z-auf '" {z-ajtYh' 
Then if we consider the function 



£?(^)=JJct,f(^-a*)-c^f'(^-a*) + ...+j^:^^^ 



d* 
where f <*^ (z) denotes j- f (5), we see that 



368 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI. 

(1) When z is replaced by {z + 2a>i), the function E{z) is increased by 

n 

n 

But S Ck is zero, since the sum of the residues o( f(z) within a period- 

Jb=l 

parallelogram is zero. Hence E{z) admits the period 2(0i. Similarly E(z) 
admits the period 2ft),. E (z) is therefore an elliptic function, with the same 
periods ea/(z), 

(2) Since the function ^^^^(z — ai) has singularities only at a* and 
congruent points, and its principal part at a* is (— 1)*** ml (z — ajb)"**""*, we see 
that E{z) has the same singularities as /(z), and the same principal parts 
at them. 

It follows from (1) and (2) that f{z)'-E{z) is a function with no 
singularities in the whole plane; and therefore, by Liouville's theorem, 
f(z) — E(z) is a constant. Thus the function f(z) can be expanded in the 
farm 

/(2r)=: Constant + 2 S J-^^' c*.f <-« (z - a,). 

This theorem may be regarded as analogous to the decomposition of a rational function 
into partial fractions, or the decomposition of a circular function into a series of co- 
tangents (§ 76). 

Example 1. Shew that 

ir sn «= f («- liT') - f («- Sir- tiT') + Constant, 
where the (-functions are formed with the periods 4 JT and 2iK\ 



1 pw rw 
1 pw r« 



Example 2. Shew that 

j 1 »(y) lf^(y) 

Extend this theorem to the case in which there are any number of variables. 

(Cambridge Mathematical Tripos, Part II, 1894.) 

212. The function a- (z). 

We shall next introduce a function ^(z)^ defined by the equation 



^log<r(^)=f(2r), 
with the condition that a (z)lz is to be unity when z=^0. 



212, 213] ELLIPTIC FUNCTIONS ; GENEBAL THEOREMS. 369 

Since the convergence of the infinite series which represents ^(z) is 
uniform, the series can be integrated term by term : we thus have 



log a iz) = log . + 2 {log (l - ^^^ ^ g^ j 



+ 



2?rwt)i + 2wa)a 2 (2ma)i + 27k»8)«j ' 



on choosing the constant of integration so as to satisfy the condition at ^ = ; 
and therefore 

the product being, as usual, extended over all integer and zero values of m 
and n, except simultaneous zeros. The absolute convergence of this product 
follows from that of the series 

1 ^^ V 2ma>i + 2na>2 / "*" 2m^iT2n6^ "*" 2 (2m«i+ 2ni»j)*j ' 
which is established by comparison with the series 

- 2 ^ 

3 (2ma>^ + 2n(o^y ' 

since the terms of the two series have ultimately a ratio of equality. 

It is evident from the product-expression that a (z) is an odd function of 
z, that its zeros are at the points z^2m(Di + 2na>s> and that z~^a'(z) tends to 
the limit unity as z tends to zero. 

The function <r (z) may be compared with the function sin 2, defined by the expansion 






d 
The relation ^ log (sin «) = cot « 

corresponds to ^ log <''(«)" f (^)* 

213. The qaasi'periodicity of the function <r {z). 
On integrating the equation 

f(^+2ah) = ?(^) + 2i;i 
we have log a{z-\- 2<o^ = log tr {z)-^ 2%^ + Constant, 

or ciz-^ 2o),) = C6*»»*^<r (z), 

where c is a constant. To determine c, write ^ = — oh ; thus 

<r (w,) = — c^"*'****^- (a)i), 
or c = — e*»»*^, 

W. A. 



24 



370 TRANSCENDENTAL FUNCTIONa [CHAP. XVL 

and therefore <r (e + 2a)i) = - 6^i («+«.) a {£). 

Similarly <r (^ + 2(0,) = - e*»«(*+«Oo- {z). 

The behaviour of the function <r {z) when its argument is increased by 
a period of (fi{z) is thus determined. By repeated application of these 
formulae we can find the value of <r (^ + 2ma)i + 2n<»9), where m and n are 
any integers. 

An example shemng how the function o- {z) may he expressed as a singly-infinite product 
We have 

<r («) =zn (1 - ^ ?-« I ea»«,+ai»«,"*'^»m«,+2«-,)« , 

the Biunmation being extended over all positive and n^ative integer and zero values uf m 
and n, exoept simultaneous zeros. This can be written in the form 

»»±i V 27»«i/ ^.Ai \ 2w«8/ 

±00 • / » \ * 

X 



m~±l jissl \ 2w«i + 2w«)2/ 

n n (1+- ^ )ea»-i+2n4./»(2m«»+2iu.J«. 

«i»±l n=l \ 2m«, + 27ki)j/ 



Now 



±00 



=.±l\ 2WKtt,/ IT 2«, 



and n n fl-5 —; — )e2«i«i+an«i4"*"H2»i4#,+amJ 



_«\ -.?«?izf 



\ 2nu»J 



^ 2mwi 



^ sm^- = %• f -2««tg 1 1 *^ 1 

= n ^<»i ^ ^—Ai 12««, (2m«,+am»,)"*"*(a«M.i+2ii^/ 

2<0j 29ta>2 

Similarly 

*« « / ^ \ ^* — +1 ^ 

n n ( 1+^5 I «a»t«i+2«Wi4 (2»n«i+2»M*a)* 

m»±i «=i \ 2mtt>i+2w«2/ 

. (2nci>a + z)ir ^ « ^ 

^ sin^-— I ^— ^ f 2w»,8 +1 '^ \ 

sr n ^<^i ^ ^««±a8ti»«, (2iiu»,+2iM.^^'(2»iii»j+aiM^«; 

-1 8in?^ 1+-^ * ' 

2a>j 2na>2 



213] 



ELLIPTIC FUNCTIONS; GENERAL THEOREMS. 



371 



Therefore 



*• 1 

2a», 



Sin 



00 

X n 

«=i 



3 . 1 ^^ sin ^ ^ ^' ^ *j / -2»M,r 



Sin 

2n<k)o7r 
Sin ~^-^- 
2a»i 



(2mM,+2i»w9) 






00 

X n 

»=i 



e 



z£ 



2a>, 



+j ^ sin 



1 



27K04S' 

sm '^ 



6 



** r 



2nMt' , . «* 



(2mM,+2n«08) (2mw,+2n«tft)' 



} 



2a) 



1 



or 



±m 1 



«*JU ~ olu — ,_ L 5 z 

2tt)i 2a>t g(2ii«,)« «=4i(2m«,+2»«,)« 

Sin*'^ 
®1 



Now write q^ e »! . 



Then 



.l„(^l^^.g!!fb±£); {,.,»-.-.g} {.. j!'-"' 



} 



. onfl»a7r 

Sin" — *- 

«1 



{ 2ltoaii r| 



m 



1-25** 008 — hg^* 

Now if the imaginary part of tojta^ is positive, we have | ^| < 1 ; and thus the infinite 
product 

1 - 2a*» cos — + ^* 

«=i (i-g^**)^ 

converges ahsolutely, since the series 

00 

n=l 

converges absolutely ; and hence we can separate off the exponential factors, and can 
write 



where (7 is a constant. 



l-2^cos-- + a*» 



«1 



The quantity (7 can be very simply determined from the relation 

«r(«+2«i)=-c^''»(*+"»)«rW; 
for this gives 






or 



(7=-5>~ 



2<k>, 



872 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI. 

We have therefore finally an expression for o- (2) as a singly-infinite product, namely 

where q=€ •*! . 

214. 2%6 integration of elliptic functions. 

The integral of any elliptic function can be found in terms of the functions 
f(«) and (r (z), by using the theorem given in §211, on the resolution of 
elliptic functions into a sum of (['-functiona 

In fact, in § 211 an expression 
has been found for the elliptic function f(z) ; the indefinite integral of this 



• • 



expression is 



c-^+ 2 c*. log o- (2: - aik) + 2 2 7 YTf <^*. f^*"*^'^^ - «*)> 

*=i *=i #=2 V*— 17' 

which is the required integral of f(z). 

Example, The expression for fp^ {t\ found by the theorem of § 211, is 



It follows that 



jii^(z)dz=ifp^(z)+ ^g^ + Constant. 



216. Expression of an elliptic furvction whose zeros and poles are knoum. 

We have already seen (§ 205) that the number of irreducible zeros of an 
elliptic function is equal to the number of its irreducible poles ; and that 
(§ 208) the sum of the aflSxes of the zeros differs from the sum of the affixes 
of the poles only by a quantity of the form {^nitoi + 2rko,), where m and n are 
integers. By replacing the zeros and poles by others congruent to them, we 
can reduce this difference to zero. Suppose this done, so that for a given 
function f{z) the irreducible zeros are Oi, a,, ... On, and the irreducible 
poles are 61, 6,, ... h^ where 

Oi + Oj + . . . + On = 61 + 61 + . . . + 6n« 

If any of the zeros or poles is multiple, of order k say, it will of course 
be counted as if it were k distinct simple zeros or poles. 

Consider now the quantity 

^ / X ^ a{z-a^)a{z'-a^) a{z-a^ 

^ a{z — b^a{Z'-b^ a{z — bf^' 



214,215] ELLIPTIC FUNCTIONS; GENERAL THEOREMS. 373 

^ We have 

Similarly J? (z + 2w^) = E (z). 

Thus E(z) is an elliptic function, with the same periods aa /{z); and 
therefore f{z)/E{z) admits these periods. 

But the function f(z)/E(z) clearly has no zeros or poles at the points 

djf eta, ... ttnj ^i> ••• Ofif 

and so has no zeros or poles at any point of the 2r-plane. Therefore, by 
Liouvilles theorem, f(z)/E{z) is a constant; and so finally 

''' ^'^''a{z^b,)a{z-'b,) o-(ir-6n)' 

where c is some constant. 

An elliptic function is therefore determinate, save for a multiplicative 
constant, when the places of its irreducible zeros and poles are known. 

This is analogous to the factorisation of a rational function : if a rational function has 
zeros at points a^ a,, ... o^, and poles at points &|, b^y ... b^, it can be expressed in the 
form 

(g-at) (g-ag) ... (z-On) 
U^-b,)(z-b^,..(z-b^y 
where c is a constant. 

Example 1. Prove that 

By differentiating this formula, shew that 

and by further differentiation obtain the addition-theorem 

J>(^+y)=-f>W-j>(y)+;{ ^g:g'g} V 

Example 2. If 

2 (aA-6A)=0, 

shew that S «^ («A- ft.) ... o- («A- 6a) ■■■ .r (ax - 6^^^ 

A=l (r(OA— a,)... 1 ...<r(ax-o») 



24—3 



374 



TRANSCENDENTAL FUNCTIONS. 



[chap. XVL 



Miscellaneous Examples. 

1. Shew that, ifp denote one of the functions an e, cnZf dn z, and if q and r denote the 
other two, it is always possible to choose constants a, 6, c, such that 



/ 



jxiz = a log (bq +cr). 



2. Shew that every elliptic function of order n can be expressed as the quotient of two 
expressions of the form 

where 6, Oj, a,* ••• ^n* ^^ constants. (Painlev^.) 

3. Prove that 

ip(^-a)if>(^-6)-if>(a-6){if>(2-a) + fr>(^-6)-i?(a)-if>(6)} 

+ j?(«)P(&). 

(Cambridge Mathematical Tripos, Part II, 1895.) 

4. Shew that 



<r (x-k-y+z) a (x-y) <r (y - z) <r {z - a;) _1 



1 i?(^) rw 

1 j?(y) r(y) 

Obtain the addition-theorem for the function fp (z) from this result 
5. Establish the identity 

1 PW i»'(^x)...««-"(^i) ^^^" ^"^ 



1 if>W jf>'W...««-*>(0 . 

where the product is taken for all int^er values of X and ft from to n, with the restriction 



6. Prove that 



f(-?-a)-f(2-6)-f(a-6) + f(2a-26) 



<r {z^2a-\-b) a {z- 2b-{-a) 



<r{2b — 2a) <r(z-a) o-(z — b)' 
(Cambridge Mathematical Tripos, Part II, 1896.) 

7. Shew that, if Zf^+Zi+Z2+z^=0y then 

{2f (z^)}'=3 {Sf (2a)} {2fp (2a)} + 2P' (2a), 

the summations being taken for X=0, 1, 2, 3. 

(Cambridge Mathematical Tripos, Part II, 1897.) 



MISC. EX8.] ELLIPTIC FUNCTIONS ; GENERAL THEOREMS. 375 

8. Prove that 

f^^ (r(Z'^Zi)(r{z-¥22)<r(z+z^)fr(z+z^) 

is a doubly-periodic function of z, such that 

= ~2a-{i(«8+28- 2,-^4)} <r{i(«3+2i-«,-«4)}(r{i(«i+28-a3-«4)}. 

(Cambridge Mathematical Tripos, Part II, 1893.) 

9. If /(z) be a doubly-periodic function of the third order, with poles at «=0i, z^Cj, 
-?=c„ and if ^ («) be a doubly-periodic function of the second order with the same periods 
and poles at 2= a, 2= /3, its value in the neighbourhood of 2= a being 

<(,(£)= A-+Xj(2-a)-|-X2(^-a)*-H..., 
prove that 

i X« {/" (a) -/" (/3)} - X {/' (a) +/' 03)} 2 * (c,) + {/(a) -/(/3)} {ZXk, + 2 <^ (c,) <^ (c,)} -0, 

(Cambridge Mathematical Tripos, Part II, 1894.) 

10. If X(;s) be an elliptic function with two poles a^ a,, and if ;?i, ^, ... z^^ be 2n 
arbitrary arguments such that 

shew that the determinant whose nth row is 

1, X(«<), X«(^<),...X-(«<), \{Zi\ \(zi)\{Zi\ X«(^)Xi(^,),...X-2(z<)Xi(z<), 

where \(^)=^X(«<), 

vanishes identically. 

(Cambridge Mathematical Tripos, Part II, 1893.) 

11. Shew that, provided certain conditions of inequality are satisfied, 

\ f\ e "I =5— (cot— --hcot-^)-h — Sg^'^sm — (»iw+»y), 

where the summation applies to all positive integer values of m and n. 

(Cambridge Mathematical Tripos, Part II, 1895.) 

12. Assuming the formula 

'W=* ' • V "" 2i, ? — (TTj^i — > 

prove that 

on condition that 

(Cambridge Mathematical Tripos, Part II, 1896.) 



376 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI. 

13. Shew that 

26 

1 



^W-6(«-6) 



(Dolbnia.) 



INDEX OF TERMS EMPLOYED. 



( The numbers refer to the pages, where the term occurs for the first time in the 

hook or is defined.) 



Absolute convergence, 12 

,, valne (modulus), 5 
Affix, 6 

Analytic function, 46 
Argand diagram, 6 
Associated Legendre functions, 281 
Asymptotic expansion, 163 
Automorphic functions, 389 

Beraoullian numbers and polynomials^ 97 
Bessel coefficients, 266 

„ functions, 274, 294 
Branch, branch-point, 66 



Circle of convergence, 29 

Coefficients, Bessel, 266 

Complex numbers, 4 

Conditions, Dirichlet's, 146 

Congruent, 325 

Contiguous, 260 

Continuation, 57 

Continuity, 41 

Contour, 47 

Convergence, 10 

absolute, 12 
circle of, 29 
„ radius of, 29 

semi-, 12 
uniform, 73 

Cosine series, 138 

Definite integral, 42 
Dependence, 40 
Derivate, 51, 53 
Determinants, infinite, 35 
Diagram, Argand, 6 
Dirichlet*8 conditions, 146 

„ integrals, 191 
Double-oircnit integrals, 258 
Doubly-periodic, 322 



♦» 



ft 



I* 



}> 



i> 



»» 



♦» 



»» 



Elliptic function, 322 

Equation, associated Legendre, 231 

Bessel, 269 

hypergeometric, 242 

Laplace's, 311 

Legendre, 206 
Essential singularity, 63 
Eulerian integrals, 184, 189 
Expansion, asymptotic, 163 
Exponents of a singularity, 245 

Fourier series, 127 

Function, analytic, 45 

associated Legendre, 231 
automorphic, 339 
Bessel, 274, 294 
Gamma-, 174 
elliptic, 322 
hypergeometric, 242 
identity of, 59 
Legendre, 209, 221 
many-valued, 66 



n 



»♦ 



II 



11 



II 



II 



II 



II 



Gamma- function, 174 
Genus, 339 
Geometric series, 13 

Hypergeometric series, 20, 240 
„ function, 242 

Identity of a function, 59 
Infinite determinants, 35 

products, 81 

series, 10 



II 



i» 



Infinity, point at, 64 

Integrals, definite, 42 

Dirichlet's, 191 
double-oircnit, 258 
Eulerian, 184, 189 

Invariants, 326 



II 



II 



II 



378 

Irreducible, 362 

Kind of Legendre functions, 209, 221 
„ Bessel „ 274, 296 

Laplace's equation, 811 

Legendre associated functions, 231 
equation, 206 
functions, 209, 221 
polynomials, 204 

Limit, 8 



INDEX. 



»i 



i» 



i» 



Many-valued function, 66 
Modulus, of complex quantity, 5 

„ Jacobian elliptic functions, 346 

Non-uniform convergence, 73 
Numbers, BernouUian, 97 
„ complex, 4 

Order of Bessel coefficients, 267 
„ functions, 274 
elliptic functions, 368 
Legendre functions, 209 

„ polynomials, 204 

pole, 63, 65 
zero, 55, 64 



>» 



a 



»» 



i» 



ti 



II 



Parallelogram, period-, 825 
Part, principal, 63 
Period, 322 

Period-parallelogram, 325 
Point, regular, 45 

,, representative, 6 

„ singular, 45 



Pole, 68, 66 

Polynomials, BernouUian, 97 

„ Legendre, 204 

Power-series, 28 
Principal part, 68 
Process of continuation, 57 
Products, infinite, 81 

Quantity, complex, 4 

Radius of convergence, 29 
Regular, 45, 46 
Residue, 83 
Representative point, 6 

Semi-convergence, 12 
Series, Fourier, 127 

geometric, 18 

hypergeometric, 20, 240 

infinite, 10 

power-, 28 

sine and cosine, 188 
Simple pole, 63 
Sine series, 138 
Singly-periodic, 322 
Singularity, 45 

„ essential, 68 

of hypergeometric equation, 245 



II 
If 
II 
II 
II 



II 



Uniform convergence, 73 
Uniformisation, 388 

Value, absolute (modulus), 5 
Zero, 65, 64 



CAMBBmOE ; PRINTED BY J. & C. V, CLAY, AT THE UMIVBBSITT PRB88. 



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